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SUEVEYING 



AND 



TRAVERSE TABLE 



.^' 



G. A. WENTWORTH, A.M. 



AUTHOR OF A SERIES OF TEXT-BOOKS IN MATHEMATICS 



REVISED EDITION 



FFR 221896 



Boston, U.S.A., and London 

GINN & COMPANY, PUBLISHERS 

1896 



V 






Entered, according to Act of Congress, in the year 1882, by 

G. A. WENTWORTH 
in the Oflfice of the Librarian of Congress, at Washington. 



Copyright, 1895, by G. A. Wentworth. 






CONTENTS. Vll 



SURVEYING. 

CHAPTER I. Definitions. Instrujients and Their Uses: 

Definitions, 193 ; instruments for measuring lines, 194 ; chaining, 
194 ; obstacles to chaining, 196 ; the surveyor's compass, 198 ; uses 
of the compass, 200 ; verniers, 203 ; the surveyor's transit, 207 ; uses 
of the transit, 208 ; the theodolite, 208 ; the railroad compass, 208 ; 
plotting, 211. 

CHAPTER II. Land Surveying: 

Determination of areas, 213 ; rectangular surveying, 217 ; field 
notes, computation, and plotting, 218 ; supplying omissions, 222 ; 
irregular boundaries, 222 ; obstructions, 222 ; modification of the 
rectangular method, 225 ; variation of the needle, 226 ; methods of 
establishing a true meridian, 227 ; dividing land, 231 ; United States 
public lands, 236 ; Burt's solar compass, 237 ; laying out the public 
lands, 239 ; Plane-table surveying, 241 ; the three-point problem, 246. 

CHAPTER III. Triangulation : 

Introductory remarks, 247 ; the measurement of base lines, 248 ; 
the measurement of angles, 249. 

CHAPTER IV. Levelling: 

Definitions, 250 ; the Y level, 251 ; the levelling-rod, 251 ; differ- 
ence of level, 252 ; levelling for section, 256 ; substitutes for the 
Y level, 260 j topographical levelling, 263. 

CHAPTER V. Railroad Surveying : 

General remarks, 264 ; cross-section work, 265 ; railroad curves, 
265. 



SUEYEYII^G. 

CHAPTER I. 

DEFINITIONS. INSTRUMENTS AND THEIR USES. 

§ 1. Definitions. 

Surveying is the art of determining and representing dis- 
tances, areas, and the relative position of points upon the 
surface of the earth. 

In plane surveying, the portion surveyed is considered as a 
plane. 

In geodetic surveying, the curvature of the earth is regarded. 

A Plumb-Line is a cord with a weight attached and freely 
suspended. 

A Vertical Line is a line having the direction of the plumb- 
line. 

A Vertical Plane is a plane embracing a vertical line. 

A Horizontal Plane is a plane perpendicular to a vertical 
line. 

A Horizontal Line is a line in a horizontal plane. 

A Horizontal Angle is an angle the sides of which are in a 
horizontal plane. 

A Vertical Angle is an angle the sides of which are in a 
vertical plane. If one side of a vertical angle is horizontal, 
and the other ascends, it is an angle of elevation ; if one side 
is horizontal, and the other descends, it is an angle of depression. 

The Magnetic Meridian is the direction which a bar-magnet 
assumes when freely supported in a horizontal position. 



194 SURVEYING. 

The Magnetic Bearing of a line is the angle it makes with 
the magnetic meridian. 

Surveying commonly comprises three distinct operations ; 
viz. : 

1. The Field Measurements, or the process of determining 
by direct measurement certain lines and angles. 

2. The Computation of the required parts from the measured 
lines and angles. 

3. The Plotting, or representing on paper the measured 
and computed parts in relative extent and position. 



THE MEASUREMENT OF LINES. 

§ 2. Instruments for Measuring Lines. 

The Gunter's Chain is generally employed in measuring land. 
It is 4 rods, or 66 feet, in length, and is divided into 100 links. 
Hence, links may be written as hundredths of a chain. 

The Engineer's Chain is employed in surveying railroads, 
canals, etc. It is 100 feet long, and is divided into 100 links. 

A Tape Measure, divided into feet and inches, is employed 
in measuring town-lots, cross-section work in railroad sur- 
veying, etc. 

In the United States Coast and Geodetic Survey, the meter 
is the unit ; and, when great accuracy is required, rods placed 
end to end, and brought to a horizontal position by means of 
a spirit-level, are employed in measuring lines. 

§ 3. Chaining. 

Eleven tally-pins of iron or steel are used in chaining ; also, 
one or more iron-shod poles called flag-staffs or range poles. 

A forward chainman, or leader, and a hind chainman, or 
follower, are required. A flag-staff having been placed at the 
farther end of the line, or at some point in the line visible 



CHAINING. 



195 



from the beginning, the follower takes one end of the chain, 
and a pin which he thrusts into the ground at the beginning 
of the line. The leader moves forward in the direction of 
the flag-staff, with the other end of the chain and the remaining 
ten pins, until the word "halt" from the follower warns him 
that he has advanced nearly the length of the chain. 

At this signal he stops, and the follower, meanwhile having 
placed his end of the chain at the beginning of the line, directs 
the leader by the words "right" or "left" until the chain is 
exactly in line with the flag-staff. This being accomplished, 
and the chain stretched tightly in a horizontal position, the 
follower says, " down." The leader then puts in a tally-pin 
exactly at the end of the chain, and answers, "down"; after 
which the follower withdraws the pin at the beginning of the 
line, and the chainmen move forward until the follower nears 
the pin set by the leader. The follower again says, " halt," 
and the operation just described is repeated. This process is 
continued until the end of the line is reached. 

If the tally-pins in the hands of the leader are exhausted 
before the end of the line is reached, when he has placed the 
last pin in the ground, he waits until the follower comes up to 
him. The follower gives the leader the ten pins in his hand, 
and records the fact that ten chains have been measured. The 
measuring then proceeds as before. If the distance from the 
last pin to the end of the line is less than a chain, the leader 
places his end of the chain at the end of the line, and the 
follower stretches tightly such a part of the chain as is 
necessary to reach to the last pin, and the number of links is 
counted. The number of whole chains is indicated by the 
number of pins in the hands of the follower, the last pin 
remaining in the ground. 

In measuring, the chain must be held in a horizontal 
position. If the ground slopes, one end of the chain must be 
raised until the horizontal position is attained. By means of 



196 SURVEYING. 

a plumb-line, or a slender staff, or, less accurately, by drop- 
ping a pin (heavy end downwards), the point vertically under 
the raised end of the chain may be determined. If the slope 
is considerable, half a chain or less may be used. 
To construct a perpendicular with a chain: 

1. When the point through which the perpendicular is to 
pass is in the line : 

Let AB (Fig. 1) represent the line, and P the point. Measure from 

P to the right or left, PC = 40 links, 

^ and place a stake at C. Let one end of 

v^y'' I the chain be held at P, and the end of 

^V'^ xj 5 the eightieth link at G ; then, taking the 

y ■ , •^j chain at the end of the thirtieth link from 

~ '• ' P, draw it so that the portions DC 

AC P B 

Pj^ - and DP are tightly stretched, and place 

a stake at D. DP will be the perpen- 
dicular required. (Why ?) 

2. When the point is without the line : 

Let AB (Fig. 1) be the line, and D the point. Take C any point in 
the line, and stretch the chain between D and C ; then, let the middle of 
the part of the chain between C and D be held in place, and swing the 
end at D around until it meets the line in P. DP will be the perpen- 
dicular required. (Why ?) 

§ 4. Obstacles to Chaining. 
1. When a tree, building, or other obstacle is encountered 
in measuring or extending a line, it may be passed by an offset 
in the following manner : 

To prolong the line AB' past a building (Fig. 2). At B erect BE per- 

^ pendicular to AB \ 

^ S- ?^ — /o"^ — ? ^ P at E erect EF per- 

. I _ ^"^V ; ! pendicular to BE ; 

E E^ F F' B,t F erect FC = BE 

^^^' ^- perpendicular to 

EF: then, CD perpendicular to FC will be in the required line, and 

AB 4- EF -\- CD = AD. By constructing two other perpendiculars, 

B'E' and P'C, the accuracy of the work will be increased. 



OBSTACLES TO CHAINING. 



197 



2. To measure across a body of water : 

Let it be required to measure the line ABCD (Fig. 3) crossing a river 
between B and C. Measure BE — 400 links ; at E erect the perpendic- 
ular EF =■ 600 links ; at B erect the perpendicular BG = 300 links. 
Place a stake at C, the intersection of AD and FG beyond the river. 




Fig. 3. 
Then BC - 400 links. For, by similar triangles, EF .BG..CE: CB. 
But EF = 2BG; hence, CE = 2 CB, and CB = BE = 400 links. EG 
and FG should be measured, in order to test the accuracy of the work. 
EG = FG = 500 liiiks. 

Instead of the above distances, any convenient distanc es may be ta ken. 
For, itEF = 2 BG, then CB = BE, and EG = FG= \JeB^ + BG^. 

3. To measure a line the end of which is invisible from the 
beginning, and intermediate points unknown : 

F 
A^ 'iG B 



E ^---^^^ ; 

Fig. 4. 
Let AB (Fig. 4) represent the line. Set up a flag-staff at D 
B and visible from A. From B let fall BC perpendicular to AD. 
ure^CandJ5C. Then 

AB 



beyond 
Meas- 



= \/aC^ + BC^. 
To find intermediate points on AB : 

At any point ^ on ^ C erect EF perpendicular to A C, and determine 
EG by the proportion AC -.CB ::AE : EG. G will be a point on AB. 
The line AD is called a Random Line. 



198 SURVEYING. 

THE MEASUREMENT OF ANGLES. 

§ 5. The Surveyor's Compass.* 

The Surveyor's Compass is shown on the following page. 

The compass circle is divided into half-degrees, and is fig- 
ured from 0° to 90° each way from the north and south points. 
In the centre of the compass circle is the pivot which supports 
the magnetic needle. The needle may be lifted from the pivot 
by a spring and pressed against the glass covering of the 
compass circle, when the instrument is not in use. The main 
plate moves around the compass circle through a small arc, 
read by the vernier, for the purpose of allowing for the varia- 
tion of the needle (§ 23). The sight standards at the extremi- 
ties of the main plate have fine slits nearly their whole length, 
with circular openings at intervals ; on the edges of the north 
standard are tangent scales for reading vertical angles ; and 
on the outside of the south standard are two eye-pieces at the 
same distance from the main plate as the zeros of the tangent 
scales, respectively. The telescopic sight (a recent improve- 
ment by the Messrs. Gurley), consists of a small telescope 
attached to the south standard. The main plate is furnished 
with two spirit levels at right angles, and turns horizontally 
upon the upper end of the hall spindle, the lower end of which 
rests in a spherical socket in the top of the tripod or Jacobs 
staff which supports the instrument. From the centre of the 
plate at the top of the tripod a plummet is suspended by 
which the centre of the compass can be placed directly over a 
definite point on the ground. 

*The instruments described on this and the following pages are 
adjusted by the maker. If they should require re-adjustment, full 
directions will he found in the manual furnished with the instruments. 

The manual published by Messrs. W. & L. E. Gurley, Troy, N. Y., 
has been freely used, by permission, in describing these instruments. 




THE SURVEYOR'S COMPASS. 



Note. The letters E and W on the face of the compass are reversed 
from their true positions. The reason for this is that if the sights are 
turned towards the west, the north end of the needle is turned towards 
the letter W, and if the north end of the needle is turned towards E, the 
sights are turned towards the east. 

If the north end of the needle poii^ts exactly towards E or W, the 
sights will range east or west. 



INSTRUMENTS AND THEIR USES. 



201 



§ 6. Uses of the Compass. 

To take the bearing* of a line. Place the instrument so that 
the plummet will be directly over one end of the line, and level 
by pressing with the hands on the main plate until the bubbles 
are brought to the middle of the spirit levels. Turn the south 
end of the instrument toward you, and sight at the flag-staff 
at the other end of the line. Eead the bearing from the north 
end of the needle. First, write N. or S. according as the north 
end of the needle is nearer N. or S. of the compass circle ; 
secondly, write the number of degrees between the north end 
of the needle and the nearest zero mark; and thirdly, write 
E. or W. according as the north end of the needle is nearer 
E. or W. of the compass circle. 

In Fig. 5 the bearing would be N. 45° W. 
In Fig. 6 the bearing would be S. 45° W. 
' In Fig. 7 the bearing would be S. 30° E. 
In Fig. 8 the bearing would be N. 60° E. 

If the needle coincides with the N.S. or E.W. line, the bear- 
ing would be N., S., E., or 
W., according as the north 
end of the needle is over 
N., S., E., or W. 

As the compass circle is 
divided into half-degrees, 
the bearing may be deter- 
mined pretty accurately to 
quarter-degrees. 

When a fence or other 
obstruction interferes with 
placing the instrument over 
the line, it may be placed 

at one side, the flag-staff fig. 7. fig. a 

being placed at an equal distance from the line. 

* The magnetic bearing is meant unless otherwise specified. 




202 SURVEYING. 

Local Disturbances. Before a bearing is recorded, care 
should be exercised that the chain, pins, and other instruments 
that would affect the direction of the needle, are removed from 
the vicinity of the compass. Even after the greatest care in 
this respect is exercised, the direction of the needle is often 
affected by iron ore, ferruginous rocks, etc. 

Reverse Bearings. When the bearing of a line has been 
taken, the instrument should be removed to the other end of 
the line and the reverse bearing taken. The number of degrees 
should be the same as before, but the letters should be reversed. 

To take the bearing of a line one end of which cannot be seen 
from the other. Eun a random line (§ 4, 3); then (Fig. 4), 

whence the angle CAB may be found. This angle combined 
with the bearing of the random line will give the 'bearing 
required. 

Another method will be given in § 19. 

To measure a horizontal angle by means of the needle. Place 
the compass over the vertex of the angle, take the bearing of 
each side separately, and combine these bearings. 

To measure angles of elevation. Bring the south end of the 
compass towards you, place the eye at the lower eye-piece, 
and with the hand hold a card on the front side of the north 
sight, so that its top edge will be at right angles to the divided 
edge and coincide with the zero mark; then, sighting over the 
top of the card, note upon a flag-staff the height cut by the 
line of sight ; move the staff up the elevation, and carry the 
card along the edge of the sight until the line of sight again 
cuts tlie same height on the staff; read off the degrees of the 
tangent scale passed over by the card. 

To measure angles of depression. Proceed in the same man- 
ner as above, using the eye-piece and tangent scale on the oppo- 
site sides of the sights, and reading from the top of the sight. 



INSTRUMENTS AND THEIR USES. 



203 



T^o = To'o of a foot. 



§ 7. Verniers. 

First form. Let AB (Fig. 9) represent a portion of a rod 
for measuring heights (§ 32). The graduation to feet and 
hundredths of a foot begins at the lower end, which rests on 
the ground when the rod is in use. The line 
extending nearly across the rod at the bottom 
of the portion shown marks the beginning of 
the fourth foot. The face of the rod is 
divided into four columns : in the first is 
written the number of feet ; in the second, 
the number of tenths ; and in the third, the 
number of hundredths. 

It is evident that, with the arrangement 
just described, heights could be measured 
only to hundredths of a foot. To enable us 
to find the height more precisely, a contri- 
vance called a Vernier is used. This is shown 
at the right of the rod. It consists of a piece 
of metal or wood, the graduated part of 
which is yy^ of a foot in length; and this 
is divided into ten equal parts. Hence, one 
division of the vernier = yL of yLy = y^^o 
of a foot ; and one division of the vernier 
exceeds one division of the rod by yii^ ~ 




The vernier slides along the face or side 
of the rod. 

To use the vernier, place the lower end of the rod upon the 
ground, and move the vernier until its index or zero mark is 
opposite the point whose distance from the ground is desired. 
In the figure, the height of the index of the vernier is evidently 
4.16 feet, increased by the distance of the index above the next 
lower line (4.16) of the rod. We shall now determine this 
distance. 



204 



SURVEYING. 



Observe which line of the vernier is exactly opposite a line 
of the rod. In this case, the line of the vernier numbered 7 
is opposite a line of the rod. Then, since each division of the 
vernier exceeds each division of the rod by j J^ ^ of a foot, 

C of the vernier is y^^o ^ of a foot above the next lower line of the rod. 
5 of the vernier is ^-^^-q of a foot above the next lower line of the rod. 



2 of the vernier is j-q^^q of a foot above the next lower line of the rod. 
1 of the vernier is xo^o o ®^ ^ ^^ot above the next lower line of the rod. 
of the vernier is j-q^^q of a foot above the next lower line of the rod. 

Hence, the required reading is 4.16 + 0.007 = 4.167 feet. 

In general, the following rule is evident: 
Move the vernier until its zero line is at the 
required height; read the height to the 7iearest 
hundredth beloio the index, and write in the 
thousandths^ place the number of the division 
line of the vernier which stands opposite any 
line of the rod. 

Second form. In this form (Fig. 10) the 
graduated part of the vernier is y§^ of a foot 
in length, and is divided into ten equal parts. 
Hence, one division of the vernier = y^ of 
j|^ = y o»^^ of a foot ; and one division of the 
vernier is less than one division of the rod 
^y tU — To%o = ToV^ of a foot. 

The height of the index of the vernier in 
Fig. 10 is 4.16 feet, increased by the distance 
of the index from the next lower line (4.16) 
of the rod. We shall now determine this 
distance. 

We observe that the line of the vernier 
numbered 7 stands exactly opposite the line 
of the rod numbered 3. H«nce, 




INSTRUMENTS AND THEIR USES. 



205 



6 of the vernier is j-^^^q of a foot above the next lower line of the rod. 
5 of the vernier is j-^^^ of a foot above the next lower line of the rod. 
4 of the vernier is j-^^qq of a foot above the next lower line of the rod. 
3 of the vernier is j-^q-q of a foot above the next lower line of the rod. 
2 of the vernier is j-q%q of a foot above the next lower line of the rod. 
1 of the vernier is jQ%jf of a foot above the next lower line of the rod. 
of the vernier is j-q\-q of a foot above the next lower line of the rod. 

Hence, the required reading is 4.16 + 0.007 = 4.167 feet ; 
and the rule is evidently the same as for the first form. 




Fig. 11. 



Compass Verniers. Let LL' (Fig. 11) represent the limb of 
the compass graduated to half-degrees, and VV the vernier 
divided into thirty equal spaces, equal to tv^enty-nine spaces 
of the limb. Then one space of the vernier is less than one 
space of the limb by 1', and the reading may be obtained to 
single minutes. 

In Fig. 11 the index or zero of the vernier stands between 
32° and 32° 30', and the line of the vernier marked 9 coincides 
with a line of the limb. Hence, the reading is 32° 9'. 

When the index moves from the zero line of the limb in a 
direction contrary to that in which the numbers of the limb 
run, the number of minutes obtained as above must be sub- 
tracted from 30' to obtain the minutes required. 

If, however, the vernier be made double, that is, if it have 
thirty spaces on each side of the zero line, it is always read 



206 



SUKVEYING. 



directly. The usual form of the double vernier, shown in 
Fig. 12, has only fifteen spaces on each side of the zero line. 
When the vernier is turned to the right less than 15' past a 
division line of the limb, read the lower figures on the left of 
the zero line at any coincidence ; if moved more than 15' past 
a division line of the limb, read the upper figures on the right 
of the zero line at any coincidence ; and vice versa. 




25 



30 



25 



20 



10 


5 z 


ero 5 


t 








mil 









__ 










^ 


z 


ero 






Fig. 12. 

Uses of the Compass Vernier. The most important use of the 
vernier of the vernier compass is in setting off the variation 
of the needle (§ 23). If the variation of the needle at any 
place is known, by means of the vernier screw the compass 
circle may be turned through an arc equal to the variation. 
If the observer stands at the south end of the instrument, the 
vernier is turned to the right or left according as the variation 
is west or east. The compass will now give the bearings of 
the lines with the time meridian. 

In order to retrace the lines of an old survey, turn the sights 
in the direction of a known line, and move the vernier until 
the needle indicates the old bearing. The arc moved over by 
the vernier will indicate the change of variation since the 
time of the old survey. If no line is definitely known, the 
rhange of variation from the time of the old survey will give 
the arc to be set off. 



instruments and their uses. 207 

§ 8. The Surveyor's Transit. 

This instrument is shown on page 209. 

The compass circle is similar to that of the compass. The 
vernier plate which carries the telescope has two verniers and 
moves entirely around the graduated limb of the main plate. 
The axis of the telescope carries a vertical circle which measures 
vertical angles to single minutes by means of a vernier. Under 
the telescope, and attached to it, is a spirit level by which hori- 
zontal lines may be run, or the difference of level between two 
stations found. The cross wires are t^^o fine fibres of spider's 
web, or fine platinum wires, which extend across the tube of 
the telescope at right angles to each other ; their intersection 
determines the optical axis or line of collimation of the tele- 
scope. The transit is levelled by four levelling screivs which 
pass through a plate firmly fastened to the ball spindle, and 
rest in depressions on the upper side of the tripod plate. 

A quick centring head enables the surveyor to change the 
position of the vertical axis horizontally without moving the 
tripod ; and a quick levelling head enables him to bring the tran- 
sit quickly to an approximately level position by the pressure 
of the hands, after which the levelling screws are used ; also, 
to change the position of the transit without changing the 
positron of the tripod legs, so as to bring the plummet exactly 
over any point. 

To level the transit by the levelling^ screws. Turn the instru- 
ment until the spirit levels on the vernier plate are parallel to 
the vertical planes passing through opposite pairs of levelling 
screws. Take hold of opposite screw heads with the thumb 
and fore-finger of each hand, and turn both thumbs in or out 
as may be necessary to raise the lower side of the parallel 
plate and lower the other until the desired correction is made. 

To use the telescope. Both the eye-piece and the object 
glass may be moved in and out by a rack-and-pinion move- 
ment. The eye-piece must be moved until the cross wires are 



208 SURVEYING. 

perfectly distinct ; then a slight movement of the eye of the 
observer, from side to side, will produce no apparent change 
in the position of the threads upon the object. The object 
glass must be moved until the object is distinctly visible; and 
this must be repeated, if the distance of the object is changed. 

§ 9. Uses of the Transit. 

The transit may be used for all the purposes indicated in 
§ 6, but with much greater precision than the compass. The 
principal use, however, of the transit is in measuring horizontal 
angles by means of the graduated limb and verniers. 

To measure a horizontal angle with the transit. Place the 
transit over the vertex of the angle ; level, and set the limb at 
zero. Turn the telescope in the direction of one of the sides of 
the angle, clamp to the spindle ; unclamp the main plate, and 
turn the telescope until it is in the direction of the other side 
of the angle, and read the angle by the verniers. The object 
of the two verniers on the vernier plate is to correct any mis- 
take that might arise from the want of exact coincidence in 
the centres of the verniers and the limb. The correct reading 
may be obtained by adding to the reading of one vernier the 
supplement of the reading of the other, and dividing by 2. 

By turning off a right angle by this method, perpendiculars 
may be constructed with greater facility than by the chain. 

§ 10. The Theodolite. 
The telescope of the transit can perform a complete revo- 
lution on its axis ; whence the name transit. The theodolite 
differs from the transit chiefly in that its telescope cannot be 
so revolved. It is not much used in this country. 

§ 11. The Kailroad Compass. 
This instrument has all the features of the ordinary com- 
pass, and has also a vernier plate and graduated limb for 
measuring horizontal angles. 




THE SURVEYOR'S TRANSIT. 



INSTRUMENTS AND THEIR USES. 



211 



§ 12. Plotting. 
The principal plotting instruments are a ruler, pencil, 
straight-line pen, hair-spring dividers, diagonal scale, a right 
triangle of wood, and a circular protractor. A T-square will 
also be found convenient. 




The Diagonal Scale. A portion of this scale is shown in 
Fig. 13. AB is the unit. AB and A'B' are divided into ten 
equal parts, and B is joined with h, the first division point to 
the left of B' ; the first division point to the left of B is 
joined with the second to the left of B', etc. 

The part of the horizontal line numbered 1 intercepted 
between BB' and Bh is evidently J^ of J^ = yi^ of the unit ; 
the part of the horizontal line numbered 2 intercepted between 
BB' and Bh is yf ^ of the unit, etc. 

The method of using this scale is as follows : 

Let it be required to lay off the distance 1.43. 

Place one foot of the dividers at the intersection of the horizontal Hne 
numbered 3 and the diagonal numbered 4, and place the other foot at 
the intersection of the vertical line numbered 1 (CC) and the horizontal 
line numbered 3 ; the distance between the feet of the dividers will be 
the distance required. For, measuring along the horizontal line num- 
bered 3, from CC to BIV is 1 ; from BB' to Bh is 0.03 ; and from Bh to 
the diagonal numbered 4 is 0.4 ; and 1 -F 0.03 -H 0.4 = 1.43. 



212 



SURVEYING. 



The Circular Protractor. This instrument (Fig. 14) usually 
consists of a semi-circular piece of brass or german silver, 
having its arc divided into degrees and its centre marked. 

To lay off an angle with the proti^actor, place the centre over 
the vertex of the angle, and make the diameter coincide with the 
given side of the angle. Mark off the number of degrees in the 
given angle, and draw a line through this point and the vertex. 




Fig. 14. 

Some protractors have an arm which carries a vernier, by 
which angles may be constructed to single minutes. 

To draw through a giuen point a line parallel to a given 
line, make one of the sides of a triangle coincide with the 
given line, and, placing a ruler against one of the other sides, 
move the triangle along the ruler until the first side passes 
through the given point ; then draw a line along this side. 

To draw through a given point a line perpendicular to a 
given line, make the hypotenuse of a right triangle coincide 
with the given line, and, placing a ruler against one of the 
other sides of the triangle, revolve the triangle about the ver- 
tex of the right angle as a centre until its other perpendicular 
side is against the ruler ; then move the triangle along the 
ruler until the hypotenuse passes through the given point, 
and draw a line along the hypotenuse. 



CHAPTEE TI. 

LAND SURVEYING. 

§ 13. Definitiox. 

Land Surveying is the art of measuring, laying out, and 
dividing land, and preparing a plot. 

§ 14. Determinatiox of Areas. 
The unit of land measure is the 

acre=: 10 square chains 
= 4 roods 
= 160 square rods, perches, or poles. 

Areas are referred to the horizontal plane, no allowance 
being made for inequalities of surface. 

For convenience of reference, the following rules for areas 
are given : 

Let A, B, and C be the angles of a triangle, and a, b, and c 
the opposite sides, respectively; and let s=^^(a-\-b-\-c). 

Area of triangle ABC = ^ base X altitude [a] 

= ^&csin^ . [b] 

J a^ sin B sin C ^ -. 

~^ sin(i^+6') ^^^ 

= ^s(s — a)(s — b)(s — c). [d] 

Area of rectangle = base X altitude. 

Area of trapezoid = ^ sum of parallel sides X altitude. 

Problem 1. To determine the area of a triangular field. 

Measure the necessary parts with a Gunter's chain, or with a chain 
and transit, and compute by formula [a], [b], [c], or [d]. 



214 



SURVEYING. 



Problem 2. To find the area of a field having any number 
of straight sides. 

{a\ Divide the field into triangles by diagonals ; find the area of each 
triangle and take the sum. 

(6) Run a diagonal, and perpendiculars from the opposite vertices to 
this diagonal. The field is thus divided into right triangles, rectangles, 
and trapezoids, the areas of w^hich may he found and the sum taken. 





Fig. 15. 



Fig. 16. 



Problem 3. To find the area of a field having an irregular 
boundary line. 

(a) Let AGBCB (Fig. 15) represent a field having a stream AEFG^ 
HKB as a boundary line. Run the line AB. From E, F, G^ H, and K, 
prominent points on the bank of the stream, let fall perpendiculars EE\ 
FF\ etc., upon AB. Regarding AE^ EF, etc., as straight lines, the por- 
tion of the field cut off by AB is divided into right triangles, rectangles, 
and trapezoids, the necessary elements of which can be measured and 
the areas computed. The sum of these areas added to the area of ABCD 
will give the area required. 

(6) When the irregular boundary line crosses the straight line joining 
its extremities, as in Fig. 16, the areas of AEFTI and HGB may be found 
separately, as in the preceding case. Then the area required = ABCD + 
IIGB-AEFH. . 



Problem 4. To determine the area of a field from two 
interior stations. 

Let ABCD (Fig. 17) represent a field, and P and P' two stations within 
it. Measure PP' with great exactness. Measure the angles between PP' 
and the Hues from P and P' to the corners of the field. 



DETERMINATION OF AREAS. 



215 



In the triangle PP'I), PP' and the angles P'PD and PP 
hence, PJD may be found. In like manner, 
PC may be found. Then in the triangle 
PDC, PD, PC, and the angle DPC are 
known ; hence, the area of PDC may be 
computed. 

In like manner, the areas of all the 
triangles about P and P' may be deter- 
mined. 

Area ABCD = PAD + PI)C + PCB 
+ PBA. Also 

Area ABCD = P'AD + P'DC + P'CB 
+ P'BA. 



D are known 




Problem 5. To determine the area of 
exterior stations. 

Let ABCD (Fig. 18) represent the field, and 
P and P' the stations. Determine the areas of 
the triangles PAD, PDC, PCB, and PBA, as 
in the preceding problem. 

Area ABCD = PAD + PDC -\- PBC - 
PBA. Also, 

Area ABCD = P'AD + P'DC + P'BA - 
P'BC. 

Exercise I. 



a field from two 




Fig. 18. 



1. Eequired the area of a triangular field whose sides are 
respectively 13, 14, and 15 chains. 

2. Eeqnired the area of a triangular field whose sides are 
respectively 20, 30, and 40 chains. 

3. Eequired the area of a triangular field whose base is 
12.60 chains, and altitude 6.40 chains. 

4. Eequired the area of a triangular field which has two sides 
4.50 and 3.70 chains, respectively, and the included angle 60°. 

5. Eequired the area of a field in the form of a trapezium, 
one of whose diagonals is 9 chains, and the two perpendiculars 
upon this diagonal from the opposite vertices 4.50 and 3.25 
chains. 



216 



SURVEYING. 




6. Kequired the area of the field ABCDEF (Fig. 19), if 
^^ = 9.25 (ih^\n.s^FF' = ^M chains, BE= 13.75 chains, DT)' 

= 7 chains, DB = 10 chains, CC = 
4 chains, and AA' = 4.75 chains. 

7. Eeo[uired the area of the field 
ABCDEF (Fig. 20), if 
AF^ = 4 chains, FF' = 6 chains, 
^'^' = 6.50 chains, AE' = 9 chains, 
AD = 14 chains, ^C" = 10 chains, 
^^'=6.50 chains, BB' = 7 chains, 
C(7'=rr 6.75 chains. 
Required the area of the field AGBCD (Fig. 15), if the 
diagonal AC = 5, BB' (the perpen- 
dicular from ^ to ^C) = l, DD' (the 
perpendicular from D to ^C) = 1.60, 
P ^^^' = 0.25, i^i<^' = 0.25, GG'== 0.60, 
I£IF = 0.52, KK' = OM, AE' = 0.2, 
E'F' = 0.50, F'G' = 0A5, G'R' = 
0.45, ^'ir'=:0.60, and ^'^ = 0.40. 
9. Required the area of the field 
AGBCD (Fig. 16), if ^D==3, ^C 
= 5, AB = 6, angle DAC = 4:5°, angle B AC = 30°, AE' = 
0.75, ^i^' = 2.25, AH = 2.53, AG' = 3.15, EE' = 0.60, FF' = 
0.40, and GG' = 0.75. 

10. Determine the area of the field ABCD from two interior 
if FP' = 1.50 chains, 
89° 35', FP'D = 349° 45', F'PD = 165° 40', 





F 


E 


4 


L 


B' [ C\ 



B 

Fig. 20. 



PPB = 3° 35', P'PC = 303° 15'. 



stations, P and P' 

PP'C = 

PP'B = 1S5°30', 
PPA = 309° 15', P'PA = 113° 45', 

11. Determine the area of the field ABCD from two exterior 
stations, P and P', if PP' = 1.50 chains, 

P'PB= 41° 10', P'Pi) = 104° 45', PP'B = 132° 15', 
P'PA= 55° 45', PP'D= 66° 45', PP'.^ = 103° 0'. 
P'PC= 77° 20', PP'C= 95° 40', 



RECTANGULAR SURVEYING. 



217 



RECTANGULAR SURVEYING. 
§ 15. Definitions. 
An East and West Line is a line perpendicular to the 
magnetic meridian. 

The Latitude of a line is the distance between the east 
and west lines through its extremities. 

The Departure of a line is the distance between the 
meridians through its extremities. 

Note. When a line extends north of the initial point the latitude is 
called a northing ; when it extends south, a southing ; when it extends 
east the departure is called an easting ; when it extends west, a westing. 

The Meridian Distance of a point is its distance from a 
meridian. 

The Double Meridian Distance of a course is double the 
distance of the middle point of the course from the meridian. 

Let AB (Fig. 21) represent a line, and NAS the magnetic 
meridian. Let BB' be perpendicular to NS. jy 

The bearing of the line AB is the angle 
BAB'. 

The latitude of the line AB is AB'. 

The departure of the line AB is BB'. 

The meridian distance of the point B is BB'. 

In the right triangle ABB', 

AB' = ABX cos BAB', 
and BB' = ABX sin BAB'. 

Hence, latitude = distance X cos of hearing, 
and departure =■ distaiice X sin of bearing. 

The latitudes and departures corresponding 
to any distance and bearing may be found 
from the above formulas by means of a table 
of natural sines and cosines, or from " The 
Traverse Table.''* fig. 21. 

* See Table VII. of Wentworth & Hill's Five-Place Logarithmic and 
Trigonometric Tables. 



218 



SURVEYING. 



§ 16. Field Notes, Computation, and Plotting. 

The field notes are kept in a book provided for the purpose. 
The page is ruled in three columns, in the first of which is 
written the number of the station ; in the second, the bearing 
of the side ; and in the third, the length of the side. 

Example 1. To survey the field ABCD (Fig. 22). 

Field Notes. 




1 


X. 20° E. 


8.66 


2 


S. 70° E. 


5.00 


3 


S. 10° E. 


10.00 


4 


N. 70° W. 


10.00 



{a) To obtain the field notes. 

Place the compass at A, the first station, 
and take the bearmg of AB (§ 6); suppose 
it to be N. 20° E. Write the result in the 
second column of the field notes opposite 
the number of the station. Measure AB 
= 8.66 chains, and write the result in the 
third column of the field notes. 

Place the compass at B, and, after testing 
the bearing of AB (§ 6), take the bearing of 
EC, measure 5C, and write the results in the 
field notes ; and so continue until the bearing 





Fig. 


- 




and len 


gth of each side havejbeer 


I recordec 


Q)) To compute 


the area. 




I. 


n. 


in. 


IV. 


V. 


W. 


VII. 


VIII. 


IX. 


X. 


XI. 


Side. 


Bearing. 


Dist. 


N. 


s. 


E. 


w. 


M.D. 


D.M.D. 


N.A. 


S.A. 


AB 


N.20°E. 


8.66 


AB' 
8.14 




BB' 

2.96 




BB' 
2.96 


BB' 

2.96 


2 ABB' 
24.0944 




BC 


S. 70° E. 


5.00 




B'C 
1.71 


C"C 
4.70 




CC 
7.66 


BB'+CC 
10.62 




2C'CBB' 
18.1602 


CD 


S. 10° E. 


10.00 




C'T>' 

9.85 


D'D 

1.74 




DD' 

9.40 


CC'+DD' 

17.06 




ID'DCC 

168.0410 


DA 


N.TO-W. 


10.00 


D'A 
3.42 




.... 


DD- 

9.40 





DD' 

9.40 


2 ADD' 

32.1480 


.... 






33.66 


11.56 


11.56 


9.40 


9.40 






56.2424 


186.2012 



FIELD NOTES. 219 

The survey may begin at any corner of the field ; hut in computing the 
area, the field notes should be arranged so that the 

' . .1, , 186.2012 

most eastern or most western station will stand 56.2424 

first. For the sake of uniformity, we shall always 2 1 129.9588 

begin with the most western station, and keep the ^^ |_64^98_sq. ch. 

^ fj .1 • 7.4 • • A -^ 6.498 acres, 
field on the right m passing around it. 

The field notes occupy the first three of the eleven columns in the 
above tablet. Columns IV. , V. , VI. , and VII. contain the latitudes and 
departures corresponding to the sides, and taken from the Traverse Table. 
The lines represented by these numbers are indicated immediately above 
each number. Column VIII. contains the meridian distances of the points 
B, C, Z), and ^, taken in order. Column IX. contains the double meridian 
distances of the courses. Their composition is indicated by the letters 
immediately above the numbers. Column X. contains the products of 
the double meridian distances by the northings in the same line. The 
first number, 

24.0944 = 2.96 X 8.14 = BB' X AB' = 2 area of the triangle ABB'; 
32.1480 = 9.40 X 3.42 = DJK x AIT = 2 area of the triangle ADIT. 
Column XI. contains the products of the double meridian distances by 
the southings in the same line. The first number, 

18.1602 = 10.62 X 1.71 = {BB' + CC) x B'C 

= 2 area of the trapezoid C'CBB'; 
168.0410 = 17.06 X 9.85 = {CC + BIY) X lyC 

— 2 area of the trapezoid lYBCC'. 
The sum of the north areas in column X. 

= 56.2424 = 2 {ABB' + ABIT). 
The sum of the south areas in column XI. 

= 186.2012 = 2 {C'CBB' + B'DCC). 
But {C'CBB' + B'DCC) - {ABB' + ABB') = ABCB. 

Hence, 2{C'CBB' + B'BCC) - 2{ABB' + ABB') = 2ABCB; 
that is, 186.2012 - 56.2424 = 129.9588 = 2 ABCB. 

Hence, area ABCB = i of 129.9588 = 64.9794 sq. ch. = 6.498 acres. 

(c) To make the plot. 

The plot or map may be drawn to any desired scale. If a line one 
inch in length in the plot represents a line one chain in length, the plot is 
said to be drawn to a scale of one chain to an inch. In this case the plot 
(Fig. 22) is drawn to a scale of eight chains to an inch. 

Draw the line NAS to represent the magnetic meridian, and lay off 
the first northing AB' = 8.14 (§ 12). Draw the indefinite line B'E per- 



220 SURVEYING. 

pendicular to NS and lay off B'B, the first easting = 2.96. Join^ and B; 
then the line AB will represent the first side of the field. Through B 
draw BC perpendicular to BB', and make BC" — 1.71, the first southing. 
Through C" draw C"C perpendicular to BC", and equal to 4.70, the 
second easting. Join B and C. The line BC will represent the second 
side of the field. 

Proceed in like manner until the field is completely represented. The 
extremity of the last line IX^, measured from JK, should fall at A. This 
will be a test of the accuracy of the plot. 

By drawing the diagonal A C, and letting fall upon it perpendiculars 
from B and D, the quadrilateral ABCD is divided into two triangles, the 
bases and altitudes of which may be measured and the area computed 
approximately. 

Other methods of plotting will suggest themselves, but the method just 
explauied is one of the best. 

Balancing the Work. 

In the survey, we pass entirely around the field ; hence, we 
move just as far north as south. Therefore, the sum of the 
northings should equal the sum of the southings. In like 
manner, the sum of the eastings should equal the sum of the 
westings. In this way the accuracy of the field work may be 
tested. 

In Example 1, the sum of the northings is equal to the sum 
of the southings, being 11.56 in each case ; and the sum of the 
eastings is equal to the sum of the westings, being 9.40 in 
each case. Hence, the work balances. 

In actual practice the work seldom balances. When it does 
not balance, corrections are generally applied to the latitudes 
and departures, by the following rules : 

The perimeter of the field : any one side 

: : total error in latitude : correction ; 
: : total error in departure : correction. 

If special difficulty has been experienced in taking a par- 
ticular bearing, or in measuring a particular line, the correc- 
tions should be applied to the corresponding latitudes and 
departures, 



FIELD NOTES. 



221 



The amount of error allowable varies in the practice of dif- 
ferent surveyors, and according to the nature of the ground. 
An error of 1 link in 8 chains would not be considered too 
great on smooth, level ground; while, on rough ground, an 
error of 1 link in 2 or 3 chains 
might be allowed. If the error 
is considerable, the field meas- 
urements should be repeated. 

Example 2. Let it be re- 
quired to survey the field AB 
CDEF (Fig. 23). 

Field Notes. 



1 


N. 73° 30' W. 


5.00 


2 


S. 16° 30' W. 


5.00 


3 


N. 28°30' W. 


7.07 


4 


N. 20° 00' E. 


11.18 


6 


S. 43°30'E. 


5.00 


6 


S. 13° 30' E. 


10.00 



243.0888 

81.4955 

2 1 161.5933 

10 1 80.7967 

8.0797 acres. 

Explanation. The first station 
in the field notes is D, but we re- 
arrange the numbers in the tablet so 
that A stands first. The northings 
and southings balance, but the east- 
ings exceed the westings by 1 link. 
We apply the correction to the west- 
ing 4.79 (the distance BE being in 
doubt), making it 4.80, and rewrite 
all the latitudes and departures in the next four columns, incor 
the correction. In practice, the corrected numbers are written in 





^ t?3 b Q to !j^ 
!^ ^ tq b Q ta 


05 
1 




t2j CQ !^ OQ CB tz| 

% s s| b b g 
g g g g g § 

^ ^ ^' H H H 


r 
1 




-J en Oi O en M 

S § g S 8 S 


7^ 






^ 


^ 
^ 


*.. * CO CO • 


93 


to 


': '. \ I I I 


ri . 


to 


05 M vl^ . 
CO tl^ -fl • 

~^ to o • 


^ 






^ 




*-tq . oC5 cots . 

: 3^ : Sb Sq : 


Co 




l^^b coO 03 to 

: : : £^b'^c^^t«' 


rn 




oshq mSj *.b ; ; ; 
i^ ^ to Ky o ; ; ; ; 


5 




_ wh3 f-tg pb r^o Mb3 


CD 




eg b q b3 
w^q ootq !^b S^ ^^ wto 

»^ tg b Q 




22 

1 

en 


1^ 








?: 

^ 


CO 




to 




CO 



porating 
red ink. 



222 



SURVEYING. 



The remainder of the computation does not require expla- 
nation. 

It will be seen that this method of computing areas is 
perfectly general. 

iV § 17. Supplying Omissions. 

If, for any reason, the bearing 
and length of any side do not appear 
in the field notes, the latitude and 
departure of this side may be found 
in the following manner : 

Find the latitudes and departures 
of the other sides as usual. The 
difference between the northings 
and southings will give the north- 
ing or southing of the unknown 
side, and the difference between 
the eastings and westings will give 
the easting or westing of the un- 
known side. 

If the length and bearing of the 
unknown side are desired, they may 
be found by solving the right tri- 
FiG. 23. angle, whose sides are the latitude 

and departure found by the method just explained, and whose 
hypotenuse is the length required. 

§ 18. Irregular Boundaries. 

If a field have irregular boundaries, its area may be found 
by offsets, as explained in § 14, Prob. 3. 

§ 19. Obstructions. 

If the end of a line be not visible from its beginning, or if 
the line be inaccessible, its length and bearing may be found 
as follows : 




OBSTRUCTIONS. 



223 



1. By means of a random line (§ 4, 3). 

2. When it is impossible to run a random line, which is 
frequently the case on account of the extent of the obstruc- 
tion, the following method may be used : 



N 



Let AB (Fig. 24) represent an inaccessible line 
whose extremities A and B only are known, and B 
invisible from A. 

Set flag-staffs at convenient points, C and D. 
Find the bearings and lengths of A C, CD, and DB, 
and then proceed to find the latitude and departure 
of AB, as in § 17. 



Fig. 24. 



Example. Suppose that we have the following notes (see 
Fig. 24): 



Side. 


Bearing. 


DIST. 


N. 


S. 


E. 


w. 


AC 
CD 
DB 


S. 45° E. 

E. 

N. 30° E. 


3.00 
3.50 
4.83 


4.18 


2.12 


2.12 
3.50 
2.42 






4.18 


2.12 


8.04 







4 1g The northing of AB is 2.06, and the easting, 8.04; which 

2.12 numbers may be entered in the tablet in the columns N. and E., 
2.06 opposite the side AB. 

If the bearing and length of AB are required, construct the 
right triangle ABC (Fig. 25), making AC = 8.04 and BC = 2.06. 



tan^^C = 



BC 2.06 



AC 8.04 
Hence, the angle BAG = 14° 22'. 
Also, 



= 0.256. 



AB = ^AC^ + BC'^ = V8.042 + 2.062 = 8.29. 
Therefore, the bearing and length of AB are N. 75° 38' E. and 8.29. 

Note. Keep all the decimal figures until the result is obtained ; then 
reject all decimal figures but two, increasing the last decimal figure 
retained by 1, if the third decimal figure is 5 or greater than 5. 



224 



SURVEYING. 



Exercise II. 

In examples 5 and 6 detours were made on account of inaccessible 
sides (§ 19, 2). The notes of the detours are written in braces. 

5. 8. 





1. 




Sta. 
1 


Bearings, 


Dist 


S. 75° E. 


6.00 


2 


S. 15° E. 


4.00 


3 


S. 75° W. 


6.93 


4 


N.45°E. 


5.00 


5 


N.45°W. 


5.19^ 



Sta. 
1 


Bearings. 


Dist. 


N.45°E. 


10.00 


2 


S. 75° E. 


11.55 


3 


S. 15° W. 


18.21 


4 


N.45°W. 


19.11 



3. 



Sta. 


Bearings. 


Dist. 


1 


N.15°E. 


3.00 


2 


N.75°E. 


6.00 


3 


S. 15° W. 


6.00 


4 


N.75°W. 


5.20 



4. 



Sta. 


Bearings. 


Dist. 


1 


N.89°45'E. 


4.91 


2 


S. 7°00'W. 


2.30 


3 


S. 28°00'E. 


1.52 


4 


S. 0°45'E. 


2.57 


5 


N.84°45'W. 


5.11 


6 


N. 2°30'W. 


5.79 



Sta. 
1 


Bearings. 


Dist. 


S. 2°15'E. 


9.68 


r 


N.51°45'W. 


2.39 


2] 


S. 85°00'W. 


6.47 




S. 55°10'W. 


1.62 


3 


N. 3°45'E. 


6.39 


4 


S. 66°45'E. 


1.70 


5 


N.15°00'E. 


4.98 


6 


S. 82°45'E. 


6.03 



6. 



sta. 


Bearings. 


Dist. 


■{ 


S. 81°20'W. 


4.28 


N.76°30'W. 


2.67 


2 


N. 5°00'E. 


8.68 


3 


S. 87°30'E. 


5.54 




S. 7°00'E. 


1.79 


4- 


S. 27°00'E. 
S. 10°30'E. 


1.94 
5.35 




N.76°45'W. 


1.70 



7. 



Sta. 
1 


Bearings. 


Dist 


N. 6°15'W. 


6.31 


2 


S. 81°50'W. 


4.06 


3 


S. 5°00/E. 


5.86 


4 


N.88°30'E. 


4.12 



Sta. 
1 


Bearings. 


Dist 


N. 5°30'W. 


6.08 


2 


S. 82°30/W. 


6.51 


3 


S. 3°00'E. 


5.33 


4 


E. 


6.72 



9. 



sta. 


Bearings. 


Dist. 


1 


N.20°00'E. 


4.62^ 


2 


N.73°00'E. 


4.16^ 


3 


S. 45°15'E. 


6.18^ 


4 


S. 38°30nV. 


8.00 


5 


Wanting. 


Wanting. 



10. 



sta. 


Bearings, 


DIat. 


1 


S. 3°00'E. 


4.23 


2 


S. 86°45'W. 


4.78 


3 


S. 37°00'W. 


2.00 


4 


N.81°00'W. 


7.45 


5 


N.61°00'W. 


2.17 


6 


N.32°00'E. 


8.68 


7 


S. 75°50'E. 


6.38 


8 


S. 14°45/W. 


0.98 


9 


S. 79°15'E. 


4.52 




rectangular method. 225 

§ 20. Modification of the Eectangular Method. 

The area of a field may be found by a modification of the 
rectangular method, if its sides and interior angles are known. 

Let A, B, C, D, represent the inte- 
rior angles of the field ABCD (Fig. / / 
26). Let the side AB determine the 
direction of reference. 

The bearing of AB, with reference 
to AB, is 0°. 

The bearing of BC, with reference 
to AB, is the angle h = 180° — B. 

The bearing of CD, with reference A^ 
to AB, is the angle c=C-b. ^'''' ^^■ 

The bearing of DA, with reference to AB, is the angle d = A. 

The area may now be computed by the rectangular method, 
regarding AB SbS the magnetic meridian. 

In practice, the exterior angles, when acute, are usually 
measured. 

As the interior angles may be measured with considerable 
accuracy by the transit, the latitudes and departures should 
be obtained by using a table of natural sines and cosines. 

Exercise III. 

1. Find the area of the field ABCD, in which the angle 
^ = 120°, B = 60°, (7=150°, and D = 30°; and the side 
^5 = 4 chains, J5C = 4 chains, CD = 6.928 chains, and Z>^ 
= 8 chains. 

Keep three decimal places, and use the Traverse Table. 

2. Find the area of the farm ABODE, in which the angle 
^ = 106° 19', ^ = 99° 40', C=120°20', i) = 86°8', and JS' = 
127° 33'; and the side ^^=79.86 rods, ^(7=121.13 rods, 
CD = 90 rods, D^= 100.65 rods, and EA = 100 rods. 

Use the table of natural sines and cosines, keeping two decimal places 
as usual. 



226 surveying. 

§ 21. General Kemarks on Determining Areas. 

Operations depending upon the reading of the magnetic 
needle must lack accuracy. Hence/ when great accuracy is 
required (which is seldom the case in land surveying), the 
rectangular method (§§ 16-19) cannot be employed. 

The best results are obtained by the methods explained in 
§§14 and 20, the horizontal angles being measured with the 
transit, and great care exercised in measuring the lines. 

§ 22. The Variation of the Needle. 

The Magnetic Declination, or variation of the needle, at 
any place, is the angle which the magnetic meridian makes 
with the true meridian, or north and south line. The variation 
is east or west, according as the north end of the needle lies 
east or west of the true meridian. Western variation is indi- 
cated by the sign -|-, and eastern by the sign — . 

Irregular Variations are sudden deflections of the needle, 
which occur without apparent cause. They are sometimes 
accompanied by auroral displays and thunder storms, and are 
most frequent in years of greatest sun-spot activity. 

Solar-Diurnal Variation. North of the equator, the north 
end of the needle moves to the west, from 8 a.m. to 1.30 p.m., 
about 6' in winter and 11' in summer, and then returns 
gradually to its normal position. 

Secular Variation is a change in the same direction for 
about a century and a half ; then in the opposite direction for 
about the same time. 

The line of no variation, or the Agonic Line, is a line join- 
ing those places at which the magnetic meridian coincides with 
the true meridian. In the United States, this line at present 
(1895) passes through Michigan, Ohio, Eastern Kentucky, 
the extreme southwest of Virginia, and the Carolinas. It is 
moving gradually westward, so that the variation is increasing 



TO ESTABLISH A TRUE MERIDIAN. 



227 



at places east of this line, and decreasing at places west of 
this line. East of this line the variation is westerly, and west 
of this line the variation is easterly. 

The table on pages 234 and 235, which has been prepared by 
permission from data furnished by the United States Coast 
and Geodetic Survey, shows the magnetic variation at different 
places in the United States and Canada for several years ; 
also, the annual change for 1895. 



§ 23. To Establish a True Meridian. 

This may be done as follows : 

1. By means of Burt'' s Sola?' Compass (§ 25). 

2. By observations of Polaris. 

The North Star or Polaris revolves about the pole at present 
at the distance of about 1^-° ; hence, it is on the meridian twice 
in 23 h. 56 m. 4 s. (a sidereal day), once above the pole, called 
the upper culmination, and 11 h. 58 m. 2 s. later below the pole, 
called the lower culmination. It attains its greatest eastern 
or western elongation, or greatest distance from the meridian, 
5 h. 59 m. 1 s. after the culmination. 

The following table gives the mean local time of the upper 
culmination of Polaris for 1895 at Washington. The time is 
growing later at the rate of about one minute in three years. 



Month. 


First Day. 


Eleventh Day. 


Twenty-first Day. 




H. M. 


H. M. 


H. M. 


January . . . 


6 35 P.M. 


5 55 P.M. 


5 16 P.M. 


February 






4 32 P.M. 


3 53 P.M. 


3 14 P.M. 


March . . 






2 42 P.M. 


2 03 P.M. 


1 23 P.M. 


April . . 






12 40 P.M. 


12 00 M. 


11 17 A.M. 


May . . . 






10 38 A.M. 


9 59 A.M. 


9 20 A.M. 


June . . . 






8 37 A.M. 


7 57 A.M. 


7 18 A.M. 


July . . . 






6 39 A.M. 


6 00 A.M. 


5 21 A. 31. 


August. . 






4 38 A.M. 


4 00 A.M. 


3 19 A.M. 


September 






2 36 A.M. 


1 57 A.M. 


1 18 A.M. 


October . 






12 39 A.M. 


11 59 P.M. 


11 20 P.M. 


November 






10 37 P.M. 


9 57 P.M. 


9 18 P.M. 


December 






8 39 P.M. 


7 59 P.M. 


7 20 P.M. 



* Polaris. ^ * * ^ 

Pole. * I 



228 SXTRVEYING. 

The time of the upper culmination of Polaris may be found 
by means of the star Mizar, which is the middle one of the 
three stars in the handle of the Dipper (in the constellation 
of the Great Bear). It crosses the meridian at almost exactly 
the same time as Polaris. Suspend from a height of about 
20 feet a plumb-line, placing the bob in a pail of water to 
lessen its vibrations. About 15 feet south of the plumb-line, 
upon a horizontal board firmly supported at a convenient 
height, place a compass sight fastened to a board a few inches 
square. At night, when Mizar by estimation approaches the 
meridian, place the compass sight in line with Polaris and the 

plumb-line, and move 
I ^ * it so as to keep it 

in this line until the 
plumb-line also falls 
on Mizar (Fig. 27). 
Note the time; then 
(1895) fifty-one sec- 
onds later Polaris will 
be on the meridian. 

This interval is 
gradually increasing 
at the rate of 21 sec- 
onds a year. 

If the lower culmination takes place at night, the time may 
be found in a similar manner. 

When Mizar cannot conveniently be used, as in the spring, 
8 Cassiopeiae may be employed. This is the star at the 
bottom of the first stroke of the W frequently imagined to 
connect roughly the five brightest stars in Cassiopeia. In 1895 
it culminates 1.75 minutes before Polaris, with an annual 
increase of the interval of 20 seconds. 

Instead of the compass sight, any upright with a small 
opening or slit may be used. 



Pole 
Polaris. ^ 



Fig. 27. 



TO ESTABLISH A TRUE MERIDIAN. 229 

{a) To locate the true meridian by the position of Polaris 
at its culmination. 

1. By using the ai^jjaratus described in finding the time of 
culmination. At tlie time of culmination bring Polaris, the 
plumb-line, and the compass sight into line. The compass 
sight and the plumb-line will give two points in the true 
meridian. 

2. By means of the transit. Bring the telescope to bear on 
Polaris at the time of culmination, holding a light near to 
make the wires visible, if the observation is made at night. 
The telescope will then lie in the plane of the meridian, 
which may be marked by bringing the telescope to a horizontal 
position. 

(h) To locate the meridian by the position of Polaris at 
greatest elongation. 

The Azimuth of a star is the angle which the meridian 
plane makes with a vertical plane passing through the star 
and the zenith of the observer. 

A star is said to be at its greatest elongation, when its 
vertical circle ZN (Fig. 28) is tangent to its diurnal circle, 
that is, perpendicular to the hour circle PN. 

Let Z (Fig. 28) represent the zenith of the place, P the pole, and N 
Polaris at its greatest elongation ; that is, when its vertical 
circle ZN is perpendicular to the hour circle PN. Let ZP^ 
ZN., and PN be arcs of great circles ; then N will be a right 
angle. 

sin PN - cos (90° - ZP) cos (90° - Z). 

[Spher. Trig. § 47.] 
But ZP— the complement of the latitude. 

Hence, 90° — ZP = the latitude of the place. 

Hence, sin PN — cos latitude x sin Z. 

sin PN 



Hence, sin Z 




cos latitude 



230 



SURVEYING. 



Hence, Z (the azimuth of Polaris) can be found if the 
latitude of the place and the greatest elongation of Polaris 
(PiV) are known. 

The following table gives the mean value of the latter 
element for each year from 1895 to 1906. 

Greatest Elongation of Polaris. 



1895 


i°i5.r 


1899 


1°13.8' 


1903 


1°12.6' 


1896 


1° 14.8' 


1900 


1°13.5' 


1904 


1°12.3' 


1897 


1°14.5' 


1901 


1° 13.2' 


1905 


1° 12.0' 


1898 


P14.r 


1902 


1°12.9' 


1906 


1°11.7' 



The greatest elongation of Polaris, or the polar distance, is 
given in the Nautical Almanac. The table gives this element 
for Jan. 1. It may be found for other dates by interpolation. 

To obtain a line in the direction of Polaris at greatest 
elongation. 

1. Bij using the apparatus for finding the time of culmina- 
tion. A few minutes before the time of greatest elongation 
(5 h. 59 m. 1 s. after culmination), place the compass sight in 
line with the plumb-line and Polaris, and keep it in line with 
these, by moving the board in the opposite direction, until the 
star begins to recede. At this moment the sight and plumb- 
line are in the required line. 

2. By means of the transit. A few minutes before the time 
of greatest elongation, bring the telescope to bear on the star, 
and follow it, keeping the v^ertical wire over the star until it 
begins to recede. The telescope will then be in the required 
line. 

To establish the meridian. Having the transit sighted in the 
direction of the line just found, turn it through an angle equal 
to the azimuth in the proper direction. 



DIVIDING LAXD. 



231 



§ 24. DiviDixG Laxd. 

The surveyor must; for tlie most part, depend on his general 
knowledge of Geometry and Trigonometry, and his own inge- 
nuity, for the solutions of problems that arise in dividing land. 

Problem 1. To divide a triangular field into two parts 
having a given ratio, by a line through a ^ 

given vertex. 

Let ABC (Fig. 29) be the triangle, and A the 
given vertex. 

7? J) 
Divide BC at D, so that -=— equals the given 
JJO 

ratio, and join A and D. ABD and ADC will 
be the parts required ; for 

ABD : ADC : : BD : DC. 

Problem 2. To cut ofi from a 
triangular field a given area, by a 
line parallel to the base. 

Let ABC (Fig. 30) be the triangle, 
and let DE be the division line re- 
quired. 

^/ABC : \/ADE ::AB: AD. 



, „ JADE 




Fig. 30. 



Problem 3. To divide a field into 
two parts having a given ratio, by a line 
through a given point in the perimeter. 

Let ABCDE (Fig. 31) represent the field, 
P the given point, and PQ the required divi- 
sion line. 

The areas of the whole field and of the 
required parts having been determined, run 
the line PD from P to a comer 2>, dividing 
the field, as near as possible, as required. 
Determine the area PBCD. 




Fig. 31. 



232 



SURVEYING. 



The triangle PDQ represents the part which must be added to PBCB 
to make the required division. 



Hence, BQ 
Note. DQ = 



Area PDQ = i X PD X DQ X sin PDQ. 
_ 2 area PDQ 
~ PDx sin PDQ 
2siYea.PDQ 



perpendicular from P on DE 
P on DE may be run and measured directly. 

Problem 4. 



This perpendicular from 




To divide a field into a given number of parts, 
so that access to a pond of water is 
given to each. 

Let ABODE (Fig. 32) represent the field, 
and P the pond. Let it be required to divide 
the field into four parts. Find the area of 
the field and of each part. 

Let AP be one division line. Run PE, 
and find the area APE. Take the difference 
between APE and the area of one of the 
required parts ; this will give the area of the 
triangle PQE, from which QE may be found, 
as in Problem 3. Join P and Q ; PA Q will 
be one of the required parts. In like manner, 
PQR and PAS are determined ; whence, 
PSR must be the fourth part required. 

Exercise IV. 

1. From the square ABCD, containing 6 a. 1 r. 24 p., part 
off 3 A. by a line EF parallel to AB. 

2. From the rectangle ABCD, containing 8 a. 1 r. 24 p., 
part off 2 A. 1 R. 32 p. by a line JSF parallel to AD = 7 ch. 
Then, from the remainder of the rectangle, part off 2 a. 3 r. 
25 P., by a line GR parallel to EB. 

3. Part off 6 A. 3 r. 12 p. from a rectangle ABCD, con- 
taining 15 A., by a line FF parallel to AB ; AD being 10 ch. 

4. From a square ABCD, whose side is 9 ch., part off a 
triangle which shall contain 2 a. 1 r. 36 p., by a line BF 
drawn from B to the side AD. 



EXAMPLES. 233 

5. From ABCD, representing a rectangle, whose length is 
12.65 ch., and breadth 7.58 ch., part off a trapezoid which 
shall contain 7 a. 3 r. 24 p., by a line BjE from B to JDC. 

6. InthetriangleJ^(7,^^ = 12ch.,^(7=10ch.,^C=8ch.; 
part off 1 A. 2 R. 16 p., below the line DU parallel to AB. 

7. In the triangle ABC, AB = 26 ch., AC == 20 ch., and 
BC=16 ch. ; part off 6 a. 1 r. 24 p., below the line DU 
parallel to AB. 

8. It is required to divide the triangular field J 5 6' among 
three persons whose claims are as the numbers 2, S, and 5, so 
that they may all have the use of a watering-place at C ; AB 
= 10 ch., ^C = 6.85 ch., and CB = 6.10 ch. 

9. Divide the five-sided field ABC HE among three persons, 
X, Y, and Z, in proportion to their claims, X paying ^500, 
Y paying $750, and Z paying $1000, so that each may have 
the use of an interior pond at P, the quality of the land being 
equal throughout. Given ^P = 8.64 ch., BC =S.27 ch., 
CB:= 8.06 ch., HE=Q>M ch., and EA = 9.90 ch. The per- 
pendicular PB upon AB = 5.60 ch., FB' upon BC= 6.08 ch., 
FD" upon C^=4.80 ch., FD'" upon HE=6A4. ch., and 
FD"" upon EA = oAO ch. Assume FR as the divisional fence 
between X's and Z's shares ; it is required to determine the 
position of the fences F3f and Pi\" between X's and Y's shares 
and Y's and Z's shares, respectively. 

10. Divide the triangular field ABC, whose sides AB, AC, 
and BC are 15, 12, and 10 ch., respectively^ into three equal 
parts, by fences EG and DE parallel to BC, without finding 
the area of the field. 

11. Divide the triangular field ABC, whose sides AB, BC, 
and AC are 22, 17, and 15 ch., respectively, among three 
persons. A, B, and C, by fences parallel to the base AB, so 
that A may have 3 a. above the line AB, B, 4 a. above A's 
share, and C, the remainder. 



234 



SURVEYING. 





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(4 




Halifax N.S. . . . 
Eastport, Me. . . . 
Bangor, Me. . . . 
Provincetown, Mass. 
Portland, Me. . . . 
Portsmouth, N.II. . 
Boston, Mass. . . . 
Cambridge, Mass. 
Quebec, Canada . . 
Providence, 11. 1. . . 
Hartford, Conn. . . 
New Haven, Conn. . 
Burlington, Vt. . . 
Williamstown, Mass. 
Montreal, Canada 
Albany, N.Y. . . . 
New York, N.Y. . . 
New Brunswick, N.J. 
Cape Henlopen, Del. 
Philadelphia, Pa. 
Cape Henry, Va. . . 
Ithaca, N.Y. . . . 
Baltimore, Md. . . 
Williamsburg, Va. . 
Harrisburg, Pa. . . 
Washington, D.C. . 
Newbern, N.C. . . 
Buffalo, N.Y. . . • 
Toronto, Canada . . 



VAEIATIOX OF THE NEEDLE. 



235 



11 


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1 




Charleston, S.C. . . 
rittsburg, I'a. . . . 

Frie, Fa 

Savannah, Ca. . . 
(Cleveland, O. . . . 
K(\y West, Fla. . . 
Detroit, Mich. . . 
Sanlt Ste. Marie, Mich. 
Cincinnati, (). . . . 
(irand Uaven, Mich. 
Nashville, Tenn. . . 
Michigan City, Ind. . 
ren.sa(M)la, Fla. . . 
Chicago, 111. . . . 
Milwaukee, Wis. . . 
Mobile, Ala. . . . 
New Orleans, La. 
St. Louis, Mo. . . . 
Duluth, Minn. . . 
(Jalveston, Tex. . . 
Omaha, Neb. . . . 
Austin, Tex. . . . 
San Antonio, Tex. . 
Denver, Col. . . . 
Salt Lake City, Utah 
San Diego, Cal. . . 
Seattle, Wash. . . 
San Francisco, Cal. . 
C. Mendocino, Cal. . 



236 SURVEYING. 

§ 25. United States Public Lands. 
Burfs Sola)' Comj^ass. 

This instrument, which is exhibited, on the following page, 
may be used for most of the purposes of a compass or transit. 
Its most important use, however, is to run north and south 
lines in laying out the public lands. 

A full description of the solar compass, with its principles, 
adjustments, and uses, forms the subject of a considerable 
volume, which should be in the hands of the surveyor who 
uses this instrument. The limits of our space will allow only 
a brief reference to its principal features. 

The main plate and standards resemble these parts of the 
compass. 

a is the latitude arc. 

b is the declination arc. 

h is an arm, on each end of which is a sola)' lens having its 
focus on a silvered plate on the other end. 

c is the hour arc. 

n is the needle-hox, which has an arc of about 36°. 

To run a north and south line with the solar compass. Set 

off the declination of the sun on the declination arc. Set oif 
the latitude of the place (which may be determined by this 
instrument) on the latitude arc. Set the instrument over the 
station, level, and turn the sights in a north and south direc- 
tion, approximately, by the needle. Turn the solar lens 
toward the sun, and bring the sun's image between the 
equatorial lines on the silvered plate. Allowance being made 
for refraction, the sights will then indicate a true north and 
gouth line, 




BURT'S SOLAR QOMPASS. 



LAYING OUT THE PUBLIC LANDS. 



239 



Laying Out the Public Lands. 

The public lands north of the Ohio Kiver and west of the 
Mississippi are generally laid out in townships approximately 
six miles square. 

A Principal Meridian, or true north and south line, is first 
run by means of Burt's Solar Compass, and then an east and 
west line, called a Base Line. 

Parallels to the base line are run at intervals of six miles, 
and north and south lines 
at the same intervals. Thus N 

the tract would be divided 
into townships exactly six 
miles square, if it were not 
for the convergence of the 

meridians on account of the * I I I I I I I ' I E 

curvature of the earth. 

The north and south 
lines, or meridians, are 
called Range Lines. The 
east and west lines, or 
parallels, are called Town- 
ship Lines. 

Let NS (Fig. 33) represent a principal meridian, and WE 
a base line ; and let the other lines represent meridians and 
parallels at intervals of six miles. 

The small squares. A, B, C, etc., will represent townships. 

A would be designated thus: T. 3 N., K. 2 W.; that is, 
township three north, range two west ; which means that the 
township is in the third tier north of the base line, and in the 
second tier west of the principal meridian. B and C, respec- 
tively, would be designated thus : T. 4 S., R. 3 W. ; and 
T. 2 N., R. 2 E. 



3 
Fig. 33. 



240 



SURTETING. 



6 
7 
18 
19 
30 
31 


5 
8 
17 

20 
29 
32 


4 
9 
16 
21 
28 
33 


3 
10 
15 

22 
27 
34 


2 
11 
14 
23 
26 
35 


1 

12 
13 
24 
25 
36 



Fig. 34. 



The townships are divided into sections approximately one 
mile square, and the sections are divided 
into quarter-sections. The township, 
section, and quarter-section corners are 
permanently marked. 

The sections are numbered, beginning 
at the northeast corner, as in Fig. 34, 
which represents a township divided 
into sections. The quarter-sections are 
designated, according to their position, 
as K E., N.W., S. E., and S. W. 

Every fifth parallel is called a Standard Parallel or Correc- 
tion Line. 

Let NS (Fig. 35) represent a principal meridian; WU a 

base line; rp, etc., meridians; 
^ and ms the fifth parallel. If 

Oj) equals six miles, mr will 
be less than six miles on 
account of the convergence 
of the meridians. Surveyors 
are instructed to make Op 
such a distance that mr shall 
be six miles; then mh, hk, 
etc., are taken similarly. At 
the correction lines north of 
,-^ ms the same operation is 
repeated. 

The township and section 
lines are surveyed in such 
an order as to throw the errors on the north and outer town- 
ships and sections. 

If, in running a line, a navigable stream or a lake more 
than one mile in length is encountered, it is meandered by 



W- 



m 


r 


h 


Jc \ 








































1 


O 


J 


P 





Fig. 35. 



PLANE-TABLE SURVEYING. 241 

marking the intersection of the line with the bank and 
running lines from this point along the bank to prominent 
points which are marked, and the lengths and bearings of the 
connecting lines recorded. 

Six principal meridians have been established and con- 
nected. In addition to these there are several independent 
meridians in the Western States and Territories which will 
in' time be connected with each other and with the eastern 
system. 

§ 26. Plane -Table Surveting."* 

After the principal lines of a survey have been determined 
and plotted, the details of the plot may be filled in by means 
of the plane-table ; or, when a plot only of a tract of land is 
desired, this instrument affords the most expeditious means 
of obtaining it. 

An approved form of the plane-table, as used in the United 
States Coast and Geodetic Survey, is shown in the plate on 
page 243. 

The Table-top is a board of well-seasoned wood, panelled 
with the grain at right angles to prevent warping, and sup- 
ported at a convenient height by a Tripod and Levelling 
Head. 

The Alidade is a ruler of brass or steel supporting a 
telescope or sight standards, whose line of sight is parallel to 
a plane perpendicular to the lower side of the ruler, and 
embracing its fiducial edge. 

The Declinatoire consists of a detached rectangular box 
containing a magnetic needle which moves over an arc of 
about 10° on each side of the point. 

* In preparing this section the writer has consulted, by permission, 
the treatise on the plane-table by Mr. E, Hergesheinier, contained in the 
report for 1880 of the U.S. Coast and Geodetic Survey. 



242 SURVEYING. 

Two spirit levels at right a^ngles are attached to the ruler or 
to the declinatoire. In some instruments these are replaced 
by a circular level, or by a detached spring level. 

The paper upon which the plot is to be made or completed 
is fastened evenly to the board by clamps, the surplus paper 
being loosely rolled under the sides of the board. 

To place the table in position. This operation, which is 
sometimes called orienting the table, consists in placing the 
table so that the lines of the plot shall be parallel to the 
corresponding lines on the ground. 

This may be accomplished approximately by turning the 
table until the needle of the declinatoire indicates the same 
bearing as at a previous station, the edge of the declinatoire 
coinciding with the same line on the paper at both stations. 

If, however, the line connecting the station at which the 
instrument is placed with another station is already plotted, 
the table may be placed in position accurately by placing it 
over the station so that the plotted line is by estimation over 
and in the direction of the line on the ground ; then making 
the edge of the ruler coincide with the plotted line, and turn- 
ing the board until the line of sight bisects the signal at the 
other end of the line on the ground. 

To plot any point. Let ah on the paper represent the line 
AB on the ground ; it is required to plot c, representing C on 
the ground. 

1. By intersection. 

Place the table in position at A (Fig. 36), plumbing a over A^ and 

p making the fiducial edge of the ruler 

:t-v^ pass through a; turn the alidade 

/ "-.^ about a until the line of sight 

/ ""v^ bisects the signal at C, and draw a 

line along the fiducial edge of the 
ruler. Place the table in position 
at JB, plumbing h over B, and repeat 
the operation just described, c will 
^ be the intersection of the two lines 
Fig. 36. thus drawn. 



-^'c 




THE PLANE-TABLE. 



tLANE-TABLE SURVEYING. 



245 




/ 

/ 

/ 


N 




\ 




/ 

/ 


"-V 




/ 

/ 


■'-^ 




b 


— a 


B 




A 



2. By resection. 

Place the table in position at A (Fig. 37), and draw a line in the direc- 
tion of C, as in the former case ; then remove the instrument to C, place 
it in position by the line drawn from 
a, make the edge of the ruler pass 
through 6, and turn the alidade about 
b until B is in the line of sight. A 
line drawn along the edge of the 
ruler will intersect the line from 
a in c. 

3. By radiation. 

Place the table in position at A 
(Fig. 38), and draw a line from a 
toward 0, as in the former cases. 
Measure A C\ and lay off ac to the >v, 

same scale as a6. ^^^ 

To plot a field ^^Ci) 

1. By radiation. 

Set up the table at any point P, 

and mark p on the paper over P. B 

Draw indefinite lines from p to- _ 

Fig 
ward A, B, C Measure PA, 

PB, , and lay off pa, pb, , to a suitable scale, and join a and 6, 

b and c, c and d, etc. 



Fig. 37. 




2. By progression. 

Set up the table at A, and draw a line from a toward B. Measure 
AB, and plot ab to a suitable scale. Set up the table in position at B, 
and in like manner determine and plot 6c, etc. 

3. By intersection. 

Plot one side as a base line. Plot the other corners by the method of 
intersection, and join. 



4. By resection. 

Plot one side as a base line. Plot the other corners by the method of 
resection, and join. 



246 



SURVEYIXG. 



The Three Point Prohlem. 

Let A, B, C represent three field stations plotted as a, h, c, 
respectively (Fig. 39) ; it is required to plot d representing a 
fourth field station Z>, visible from A, B, and C. 




Fig. 39. 

Place the table over D, level and orient approximately by 
the declinatoire. Determine d by resection as follows : Make 
the edge of the ruler pass through a and lie in the direction 
aA, and draw a line along the edge of the ruler. In like 
manner, draw lines through b toward B and through c toward 
C. If the table were oriented perfectly these lines would 
meet at the required point d, but ordinarily they will form 
the triangle of error ^ ah, ac, he. In this case, through a, b, and 
ab', a, G, and ac; and b, c, and be, respectively, draw circles; 
these circles will intersect in the required point d. Eor at 
the required point the sides ab, ae, be must subtend the same 
angle as at the points ab, ae, be, respectively. Hence, the 
required point d lies at the intersection of the three circles 
mentioned. The plane-table may now be oriented accurately. 

Note. The three point problem may be solved by fastening on the 
board a piece of tracing paper and marking the point d representing J), 
after which lines are drawn from d toward A, B, and C. The tracing 
paper is then moved until the lines thus drawn pass through a, 6, c, 
respectively, when by pricking through d the point is determined on the 
plot below. 



CHAPTER III. 

TRIANGULATION.* 

§ 27. Introductory Remarks. 

Geographical positions upon tlie surface of the earth are 
commonly determined by systems of triangles which con- 
nect a carefully determined base line with the points to be 
located. 

Let F (Fig. 40) represent a point whose position with refer- 
ence to the base line AB is 
required. Connect AB with 
F by the series of triangles 
ABC, ACD, ADE, and DBF, 
so that a signal at C is visible 
from A and B, a signal at D 
visible from A and C, a signal ^^^- ^• 

at E visible from A and D, and a signal at F visible from 
D and E. In the triangle ABC, the side AB is known, 
and the angles at A and B may be measured ; hence, AC 
may be computed. In the triangle ACD, AC i^ known, 
and the angles at A and C may be measured ; hence, AD 
may be computed. In like manner DE and EF or DF 
may be determined. DF, or some suitable line connected 
with DF, may be measured, and this result compared with 
the computed value to test the accuracy of the field meas- 
urements. 

* In preparing this chapter the writer has consulted, by permission, 
recent reports of the United States Coast and Geodetic Survey. 




248 SURVEYING. 

Three orders of triangulation are recognized, viz. : 

Primary, in which, the sides are from 20 to 150 miles in 
length. 

Secondary, in which the sides are from 5 to 40 miles in 
length, and which connect the primary with the tertiary. 

Tertiary, in which the sides are seldom over 5 miles in 
length, and which bring the survey down to such dimensions 
as to admit of the minor details being filled in by the compass 
and plane-table. 

§ 28. The Measurement of Base Lines. 

Base lines should be measured with a degree of accuracy 
corresponding to their importance. 

Suitable ground must be selected and cleared of all obstruc- 
tions. Each extremity of the line may be marked by cross 
lines on the head of a copper tack driven into a stub which is 
sunk to the surface of the ground. Poles are set up in line 
about half a mile apart, the alignment being controlled by a 
transit placed over one end of the line. 

The 'preliminary measurement may be made with an iron 
wire about one-eighth of an inch in diameter and 60™ in 
length. In measuring, the wire i's brought into line by means 
of a transit set up in line not more than one-fourth of a mile 
in the rear. The end of each 60°" is marked with pencil lines 
on a wooden bench whose legs are thrust into the ground after 
its position has been approximately determined. If the last 
measurement exceeds or falls short of the extremity of the 
line, the difference may be measured with the 20"" chain. 

The final meas%irement is made with the base apparatus^ 
which consists of bars 6"^ long, which are supported upon 
trestles when in use. These bars are placed end to end, and 
brought to a horizontal position, if this can be quickly accom- 
plished ; if not, the angle of inclination is taken by a sector, 
or a vertical offset is measured with the aid of a transit, so 
that the exact horizontal distance can be computed. 



MEASUREMENT OF ANGLES. 249 

A thermometer is attached to each bar, so that the tempera- 
ture of the bar may be noted and a correction for temperature 
applied. 

The method of measuring lines varies according to the 
required degree of accuracy in any particular case, but the 
brief description given above will give the student a general 
idea of the methods employed. 

§ 29. The Measurement of Angles. 

Angles are measured by the transit with much greater accu- 
racy than by the compass, since the reading of the plates of 
the transit is taken to minutes, and by means of microscopes 
to seconds, while the reading of the needle of the compass is 
to quarter or half-quarter degrees. 

In order to eliminate errors of observation, and errors 
arising from imperfect graduation of the circles, a large 
number of readings is made and their mean taken. Two 
methods are in use ; viz., repetition and series. 

The method of repetition consists, essentially, in measuring 
the angles about a point singly, then taking two adjacent 
angles as a single angle, then three, etc. ; thus " closing the 
horizon," or measuring the whole angular magnitude about a 
point in several different ways. 

The method of series consists, essentially, in taking the 
readings of an angle with the circle or limb of the transit 
in one position, then turning the circle through an arc and 
taking the readings of the same circle again, etc. ; thus read- 
ing the angle from successive portions of the graduated circle. 

On account of the curvature of the earth, the sum of the 
three angles of a triangle upon its surface exceeds 180°. This 
spherical excess, as it is called, becomes appreciable only when 
the sides of the triangle are about 5 miles in length. To 
determine the angles of the rectilinear triangle having the 
same vertices, one-third of the spherical excess is deducted 
from each spherical angle. 



CHAPTER IV. 



LEVELLING. 



§ 30. Definitions, Curvature, and Refraction. 

A Level Surface is a surface parallel with the surface of 
still water ; and is, therefore, slightly curved, owing to the 
spheroidal shape of the earth. 

A Level Line is a line in a level surface. 
Levelling is the process of finding the difference of level of 
two places, or the distance of one place above or below a level 
line through another place. 

The Line of Apparent Level of a place is a tangent to the 
level line at that place. Hence, the line of apparent level is 
perpendicular to the plumb-line. 

The Correction for Curvature is the deviation of the line of 
apparent level from the level line for any distance. 

Let t (Fig. 41) represent the line of apparent level of the 
place P, a the level line, d the diame- 
ter of the earth ; then c represents the 
correction for curvature. To compute 
the correction for curvature : 

^2=.c(c+f^).(Geom.§348.) 

Therefore, c = — ; — ; = — 
G-\-d d 

approximately, since c is very small 

compared with d. and t = a without 
Fig. 41. . , , 

appreciable error. 

Since d is constant (= 7920 miles, nearly), the correction for 

curvature varies as the square of the distance. 




THE LEVELLING ROD. 



251 



81 



Example. What is the correction for curvature for 1 mile ? 
By substituting in the formula deduced above, 
a^ 1^ . „ . 
^=^ = 7920^^'^^ ^^- 
Hence, the correction for curvature for any 
distance may be found in inches, approximately, 
by multiplying 8 by the square of the distance 
expressed in miles. 

Note. The effect of curvature is to make an object 
appear lower than it really is ; and the effect of refraction 
of light, caused by the greater density of the atmosphere 
near the surface of the earth, is to make an object appear 
higher than it really is. AVhen both effects are taken into 

account c is more correctly expressed by c = f of — • 

§ 31. The Y Level. 

This instrument is shown on page 253. 

The telescope is about 20 inches in length, and 
rests on supports called F's, from their shape. 
The spirit level is underneath the telescope, and 
attached to it. The levelling-head and tripod are 
similar to the same parts of the transit. 

§ 32. The Levelling Rod. 

The two ends of the Philadelphia levelling rod 
are shown in Fig. 42. The rod is made of two 
pieces of wood, sliding upon each other, and held 
together in any position by a clamp. 

The front surface of the rod is graduated to 
hundredths of a foot up to 7 feet. If a greater 
height than 7 feet is desired, the back part of the 
rod is moved up until the target is at the required 
height. When the rod is extended to full length, 
the front surface of the rear half reads from 7 to 
13 feet, so that the rod becomes a self -reading rod 13 feet long. 



|gS 



Fig. 42. 



252 



SURVEYING. 



The target slides along the front of the rod, and is held in 
place by two springs which press upon the sides of the rod. 
It has a square opening at the centre, through which the 
division line of the rod opposite to the horizontal line of the 
target may be seen. It carries a vernier by which heights 
may be read to thousandths of a foot (§ 7). 

§ 33. Difference of Level. 

To find the difference of level between two places visible from 
an intermediate place, when the difference of level does not 
exceed 13 feet. 

Let A and B (Fig. 43) represent the two places. Set the 
Y level at a station equally distant, or nearly so, from A and 




Fig. 43. 



B, but not necessarily on the line AB. Place the legs of the 
tripod firmly in the ground, and level over each opposite pair 
of levelling screws, successively. Let the rodman hold the 
levelling rod vertically at A. Bring the telescope to bear 
upon the rod (§ 8), and by signal direct the rodman to move 
the target until its horizontal line is in the line of apparent 
level of the telescope. Let the rodman now record the height 
AA' of the target. In like manner find BB'. The difference 
between AA' and BB' will be the difference of level required. 
If the instrument is equally distant from A and Bj or nearly 
so, the curvature and the refraction on the two sides of the 
instrument balance, and no correction for curvature or refrac- 
tion will be necessary. 



ilTT"^^^ '-W I " ' H III ^^ 




THE Y LEVEL. 



DIFFERENCE OF LEVEL. 



255 



If the instrument be set up at one station, and the rod at 
the other, the difference between the heights of the optical 
axis of the telescope and the target, corrected for curvature 
and refraction, will be the difference of level required. 

To find the difierence of level of two places, one of which 
cannot be seen from the other, and both invisible from the same 
place ; or, when the two places difier considerably in level. 

Let A and D (Fig. 44) represent the two places. Place the 
level midway between A and some intermediate station B. 




Fig. 44. 



Find AA' and BB', as in the preceding case, and record the 
former as a back-sight and the latter as a fore-sight. Select 
another intermediate station C, and in like manner find the 
back-sight BB" and the fore-sight CC ] and so continue until 
the place D is reached. 

The difference between the sum of the fore-sights and the sum 
of the back-sights will be the difference of level required. 
For, the sum of the fore-sights 

= BB'-\-CC^-{-DD' 
= BB" -^ B'B" -\- CC" -{- C'C" -\- BD'. 
The sum of the back-sights 

= AA'-\-BB"-\-CC". 
Hence, the difference = B'B" + C"(7" + BB' — AA' 
= AA"-AA' = AA". 



256 surveying. 

§ 34. Levelling for Section. 

The intersection of a vertical plane with the surface of the 
earth is called a Section or Profile. The term profile, how- 
ever, usually designates the Plot or representation of the 
section on paper. 

Levelling for Section is levelling to obtain the data neces- 
sary for making a profile or plot of any required section. 

A profile is made for the purpose of exhibiting in a single 
view the inequalities of the surface of the ground for great 
distances along the line of some proposed improvement, such 
as a railroad, canal, or ditch, and thus facilitating the estab- 
lishment, of the proper grades. 

The data necessary for making a profile of any required 
section are, the heights of its different points above some 
assumed horizontal plane, called the Datum Plane, together 
with their horizontal distances apart or from the beginning of 
the section. 

The position of the datum plane is fixed with reference to 
some permanent object near the beginning of the section, 
called a Bench Mark, and, in order to avoid negative heights, 
is assumed at such a distance below this mark that all the 
points of the section shall be above it. 

The heights of the different points of the section above 
the datum plane are determined by means of the level and 
levelling-rod ; and the horizontal length of the section is 
measured with an engineer's chain or tape, and divided into 
equal parts, one hundred feet in length, called Stations, 
marked by stakes numbered 0, 1, 2, 3, etc. 

Where the ground is very irregular, it may be necessary, 
besides taking sights at the regular stakes, to take occasional 
sights at points between them. If, for instance*, at a point 
sixty feet in advance of stake 8 there is a sudden rise or fall 
in the surface, the height of this point would be determined 
and recorded as at stake 8.60. 



LEVELLING FOR SECTION. 



257 



The readings of the rod are ordinarily taken to the nearest 
tenth of a foot, except on hench marks and points called 
turning 2^oints, where they are taken 
to thousandths of a foot. 

A Turning Point is a point on 
which the last sight is taken just 
before changing the position of the 
level, and the first sight from the 
new position of the instrument. A 
turning point may be coincident with 
one of the stakes, but must always 
be a hard point, so that the foot of 
the rod may stand at the same level 
for both readings. 

To explain the method of obtain- 
ing the field notes necessary for 

making a profile, let 0, 1, 2, 3, 

11 (Fig. 45) represent a portion of a 
section to be levelled and plotted. 
Establish a bench mark at or near 
the beginning of the line, measure 
the horizontal length of the section, 
and set stakes one hundred feet apart, 
numbering them 0, 1, 2, 3, etc. Place 
the level at some point, as between 2 
and 3, and take the reading of the rod 
on the bench = 4.832. Let PP^ rep- 
resent the datum plane, say 15 feet 
below the bench mark, then 

15 + 4.832 = 19.832 

will be the height of the line of sight 
AB, called the Height of the Instru- 
ment, above the datum plane. Now take the reading at 
= 5.2 = 0^, and subtract the same from 19.832, whicli 




258 SURVEYING. 

leaves 14.6 = OP, the height of the point above the datum 
plane. Next take sights at 1, 2, 3, 3.40, and 4 equal respec- 
tively to 3.7, 3.0, 5.1, 4.8, and 8.3, and subtract the same 
from 19.832 ; the remainders 16.1, 16.8, 14.7, 15.0, and 11.5 
will be the respective heights of the points 1, 2, 3, 3.40, 
and 4. Then, as it will be necessary to change the position 
of the instrument, select a point in the neighborhood of 4 
suitable as a turning point (t.p. in the figure), and take a 
careful reading on it = 8.480 ; subtract this from 19.832, and 
the remainder, 11.352, will be the height of the turning point. 
Now carry the instrument forward to a new position, as 
between 5 and 6, shown in the figure, while the rodman 
remains at t.p. ; take a second reading on t.p. = 4.102, and 
add it to 11.352, the height of t.p. above FF' ; the sum 15.454 
will be the height of the instrument CD in its new position. 
Now take sight upon 5, 6, and 7, equal respectively to 4.9, 
2.8, and 0.904; subtract these sights from 15.454, and the 
results 10.6, 12.7, and 14.550 will be the heights of the points 
5, 6, and 7 respectively. The point 7, being suitable, is made 
a turning point, and the instrument is moved forward to a 
point between 9 and 10. The sight at 7 = 6.870 added to 
the height of 7 gives 21.420 as the height of the instrument 
EF in its new position. The readings at 8, 9, 10, and 11, 
which are respectively 5.4, 3.6, 5.8, and 9.0, subtracted from 
21.420, will give the heights of these points, namely, 16.0, 
17.8, 15.6, and 12.4. 

Proceed in like manner until the entire section is levelled, 
establishing bench marks at intervals along the line to serve 
as reference points for future operations. 

As wind and bright sunshine affect the accuracy of levelling, 
for careful work a calm and cloudy day should be chosen ; and 
great pains be taken to hold the rod vertical and to manipu- 
late the level properly. 

A record of the work described above is kept as follows .• 



LEVELLING FOR SECTION. 



259 



Station. 


+ s. 


H.I. 


-s. 


H.S. 


Remabks. 


B 


4.832 






15. 


Bench on rock 20 ft. 







19.832 


5.2 


14.6 


south of 0. 


1 






3.7 


16.1 




2 






3.0 


16.8 




3 






5.1 


14.7 


3 to 3.40 turnpike road. 


3.40 






4.8 


15.0 




4 






8.3 


11.5 




tp. 


4.102 




8.480 


11.352 




5 




15.454 


4.9 


10.6 




6 






2.8 


12.7 




7 


6.870 




0.904 


14.550 




8 




21.420 


5.4 


16.0 




9 






3.6 


17.8 




10 






5.8 


15.6 




11 






9.0 


12.4 




B 










Bench on oak stump 


12 










27 ft. N.E. of 12, 


etc. 










etc. 



The first column contains the numbers or names of all the 
points on which sights are taken. The second column con- 
tains the sight taken on the first bench mark, and the sight on 
each turning point taken immediately after the instrument 
has been moved to a new position. These are called Plus 
Sights {-\- S.) because they are added to the heights of the 
points on which they are taken to obtain the height of the 
instrument given in the third column (H.I.). The fourth 
column contains all the readings except those recorded in the 
second column. These are called Minus Sights (— S.) because 
they are subtracted from the numbers in the third column to 
obtain all the numbers in the fifth column except the first, 
which is the assumed depth of the datum plane below the bench. 
The fifth column (H.S., height of surface) contains the required 
heights of all the points of the section named in the first column 
together with the heights of all benches and turning points. 



260 SURVEYING. 

To find the difference of level between any two points of 
the section, we have only to take the difference between the 
numbers in the fifth column opposite these points. 

The real field notes are contained in the first, second, fourth, 
and last columns ; the other columns may be filled after the 
field operations are completed. The field book may contain 
other columns; one for height of grade (H.G.), another for 
depth of cut (C.) and another for height of embankment or 
fill {F.). 

To plot the section. Draw a line PP' (Fig. 45), to repre- 
sent the datum plane, and beginning at some point as F, lay 
off the distances 100, 200, 300, 340, 400 feet, etc., to the 
right, using some convenient scale, say 200 feet to the inch. 
At these points of division erect perpendiculars equal in 
length to the height of the points 0, 1, 2, 3.40, 4, etc., 
given in the fifth column of the above field notes, using in 
this case a larger scale, say 20 feet to the inch. Through the 
extremities of these perpendiculars draw the irregular line 

0, 1, 2, 3 11, and the result, with some explanatory figures, 

will be the required plot or profile. 

The making of a profile is much simplified by the use of 
profile paper ^ which may be had by the yard or roll. 

If a horizontal plot is required, the bearings of the different 
portions of the section must be taken. 

A plot should be made, if it will assist in properly under- 
standing the field work, or if it is desirable for future reference 
in connection with the field notes. 

§ 35. Substitutes for the Y Level. 

For many purposes not requiring accuracy, the following 
simple instruments in connection with a graduated rod will be 
found sufficient. 

The Plumb Level (Fig. 46) consists of two pieces of wood 
joined at riglit angles. A straight line is drawn on the 



SUBSTITUTES FOR THE Y LEVEL. 



261 



upright perpendicular to the upper edge of the cross- 
head. 

The instrument is fastened to a support by a screw through 
the centre of the cross-head. The upper edge of the cross- 
head is brought to a level by making the line on the upright 
coincide with a plumb-line. 



Fig. 46. 



-feEE-EEi 



Fig. 47. 



Fig. 48. 



A modified form is shown in Fig. 47. A carpenter's square 
is supported by a post, the top of which is split or sawed so as 
to receive the longer arm. The shorter arm is made vertical 
by a plumb-line which brings the longer arm to a level. 

The Water Level is shown in Fig. 48. The upright tubes 
are of glass, cemented into a connecting tube of any suitable 
material. The whole is nearly filled with water, and sup- 
ported at a convenient height. The surface of the water in 
the uprights determines the level. 

By sighting along the upper surface of the block in which 
the Spirit Level is mounted for the use of mechanics, a level 
line may be obtained. 



Exercise V. 

1. Find the difference of level of two places from the fol- 
lowing field notes : back-sights, 5.2, 6.8, and 4.0 ; fore-sights, 
8.1, 9.5, and 7.9. 

2. Write the proper numbers in the third and fifth columns 
of the following table of field notes, and make a profile of the 
section : 



262 



SURVEYING. 



Station. 


+ s. 


H.I. 


-s. 


H.S. 


Remakks. 


B 


0.944 






20 


Bench on post 22 ft. 









7.4 




north of 0. 


1 






5.6 






2 






3.9 






3 






4.6 






tp. 


3.855 




5.513 




4 


4 






4.9 






5 






3.5 






6 






1.2 







•3. Stake of the following notes stands at the lowest point 
of a pond to be drained into a creek; stake 10 stands at the 
edge of the bank, and 10.25 at the bottom of the creek. Make 
a profile, draw the grade line through and 10.25, and fill 
out the columns H. G. and C, the former to show the height 
of grade line above the datum, and the latter, the depth of 
cut at the several stakes necessary to construct the drain. 



Station. 


+ S. 


H.I. 


-s. 


H.S. 


H.G. 


c. 


Remarks. 


B 


6.000 






25 






Bench on rock 









10.2 




20.8 


0.0 


30 feet west of 


1 






5.3 






5.3 


stake 1. 


2 






4.6 










3 






4.0 










4 






6.8 










5 


4.572 




7.090 










6 






3.9 










7 






2.0 










8 






4.9 










9 






4.3 










10 






4.5 










10.25 






11.8 











Horizontal scale, 2 ch. = 1 in. 
Vertical scale, 20 ft. = 1 in. 



TOPOGRAPHICAL LEVELLING. 



263 



§ 36. Topographical Levelling. 

The principal object of topographical surveying is to show 
the contour of the ground. This operation, called topographi- 
cal levelling, is performed by representing on paper the curved 
lines in which parallel horizontal planes at uniform distances 
apart would meet the surface. 

It is evident that all points in the intersection of a horizontal 
plane with the surface of the ground are at the same level. 
Hence, it is only necessary to find points at the same level, 
and join these to determine a line of intersection. 

The method commonly employed will be understood by a 
reference to Fig. 49. The ground ABCD is divided into 
equal squares, and a numbered stake driven at each intersec- 
tion. By means of a level and levelling rod the heights of the 
other stations above m and D, the lowest stations, are deter- 
mined. A plot of the ground with the intersecting lines is 
then drawn, and the height of each 
station written as in the figure. 

Suppose that the horizontal 
planes are 2 feet apart; if the 
first passes through m and D, 
the second will pass through p, 
which is 2 feet above m ; and 
since n is Z feet above m, the 
second plane will cut the line 
mn in a point s determined by 
the proportion mn : ms : : 3 : 2. 
In like manner the points t, q, 
and r are determined. 

The irregular line tsp qr represents the intersection of 

the second horizontal plane with the surface of the ground. 
In like manner the intersections of the planes, respectively, 
4, 6, and 8 feet above m are traced. The more rapid the change 
in level the nearer these lines will approach each other. 




CHAPTEE V. 

RAILROAD SURVEYING. 

§ 37. General Remarks. 

When the general route of a railroad has been "determined, 
a middle surface line is run with the transit. A profile of this 
line is determined, as in § 34. The levelling stations are com- 
monly 1 chain (100 feet) apart. Places of different level are 
connected by a gradient line, which intersects the perpendic- 
ulars to the datum line at the levelling stations in points 
determined by simple proportion. Hence, the distance of each 
levelling station, above or below the level or gradient line 
which represents the position of the road bed, is known. 



§ 38. Cross Section Work. 
G' f ly 




Excavations. If the road bed lies below the surface, an 
excavation is made. 

Let A CDB (Fig. 50) represent a cross section of an excava- 
tion, / a point in the middle surface line, /' the corresponding 
point in the road bed, and CD the width of the excavation at 
the bottom. The slopes at the sides are commonly made so 
that AA' = iA'C, and BB' = ^DB'. ff and CD being known, 
the points A, B, C, and D' are readily determine'd by a level 
and tape measure. 



RAILROAD CURVES. 



265 



If from the area of the trapezoid ABB' A' the areas of the 
triangles AA'C and BB'I) be deducted, the remainder will be 
the area of the cross section. 

In like manner the cross section at the next station may be 
determined. These two cross sections will be the bases of a 
frustum of a quadrangular pyramid whose volume will be the 
amount of the excavation, approximately. 

Embankments. If the road bed lies above the surface, an 
embankment is made, the cross section of which is like that of 
the excavation, but inverted. 




Fig. 51 represents the cross section of an embankment 
which is lettered so as to show its relation to Fig. 50. 



39. Eailroad Curves. 



When it is necessary to change the direction of a railroad, 
it is done gradually by a 
curve, usually the arc of 
a circle. 

Let AF and AO (Fig. 
52) represent two lines to 
be thus connected. Take 
any convenient distance 
AB = AE= t The inter- 
section of the perpendicu- 
lars BC and EC deter- 
mines the centre C, and the radius of curvature BC =^r. 
The length of the radius of curvature will depend on 




266 



SURVEYING. 



the angle A and the tangent AB. For, in the right 
triangle ABC, 

tan BAC = -r^ > or tan ^A = -• 
Hence, r=^t tan ^A. 

The degree of a railroad curve is the angle subtended at the 
centre of the curve by a chord of 100 feet. If D is the degree 
of a curve and r its radius, 

50 
sin^i) = — and r = 50csc-^i>. 

For example, a 6° curve has a radius of 955.37 feet. 



To Lay out the Curve. 

First Method. Let Bm (Fig. 53) represent a portion of the 
tangent. It is required to find mP, the 
perpendicular to the tangent meeting 
the curve at P. 

mP = Bn=^CB—Cn. 
But CD = r. 




and 



Cn = ^'CP'-Pn 




Fig. 54. 



Hence, mP = r — V/^ — t^. 

Second Method. It is required to find 
mP (Fig. 54) in the direction of the 
centre. 

mP = mC — PC. 

But mC='^~BC''-\-lBm=-J^i^^. 
Hence, 

mP = VrM^ — n 



RAILROAD CURVES. 



267 




Fig. 55. 



Third Method. Place transits afc B and E (Fig. 55). Direct 
the telescope of the former 
to E, and of the latter to A. 
Turn each toward the curve 
the same number of degrees, 
and mark P, the jjoint of 
intersection of the lines of 
sight. P will be a point in 
the circle to which AB and 
AE are tangents at B and E, respectively. 

Fourth Method. If the degree D of the curve is given and 
the tangent BA at B (Fig. 
56), place the transit at B 
and direct toward A. Turn 
off successively the angles 

ABP, PBP', P'BP'\ each 

equal to -J-Z), and take DP, 

PP\ P'P", each 100 ft., 

the length of the tape. Then 

P, P', P", lie on the 

required curve. 

If the angle A and the tangent distance BA = t are given, 
D can be found from the formulas 




Fia. 56. 



sin^i>== — J r = ttain^A, 



50 
'. sin|-i>^ — cot -J- J. 




TRANSIT WITH SOLAR ATTACHMENT. 



The circles shown in the cut are intended to represent in miniature 
circles supposed to be drawn upon the concave surface of the heavens. 



SURVEYING. 



1. 8 A. 64 p. 

2. 29 A. 7f p. 

3. 4 A. 5^^ P. 

4. 115^ p. 



1. 2 A. 26 p. 

2. 20 A. 12 p. 

3. 2 A. 54 p. 

4. 2 A. 151 p. 





Exercise 


I. 






5. 


3 A. 78 p. 




9. 


13.0735, 


6. 


13 A. 6rV p. 




10. 


2 A. 58^ p. 


7. 


11 A. 157 p. 




11. 


4 A. 35 p. 


8. 


7.51925. 
Exercise 


II. 






5. 


8 A. 54 p. 




8. 


3 A. 122 p. 


6. 


5 A. 42 p. 




9. 


6 A. 2 p. 


7. 


2 A. 78 p. 




10. 


9 A. 40 p. 



Exercise III. 
1. 2 A. 12^ p. 2. 98 A. 92 p. 

Exercise IV. 

1. AE = 3.75 ch. 9. EM {on EA) = 2.5087 ch. 

2. AE = 3.50 ch. ; AN (on AB) = 6.439 ch. 

10. LetEQ>DF, 



EG = 3.42 ch. 
AE = 4.55 ch. 



rAE = 12.247 ch. 

4. ^£' = 5.50ch. \aG= 9.798 ch. 

5. CE = 4.456 ch. *^^^ 1 AD = 8.659 ch. 

6. AD = 2.27 bch.; [ AF = 6.928 ch. 
BE = 1.S2 ch. 11. Let DG > EF, 

7. AD = 4.51 ch. ; rCG= 14.862 ch. 
ii£: = 3.61ch. I Ci)= 13.113 ch. 

8. The distances on AB are 2, 3, ^^^^ ] CF = 11.404 ch. 
and5ch. I C^= 10.062 ch. 



ANSWERS. 



25 



Exercise V. 

1. 9.5 feet. 

2. Third column : 26.944 opposite ; 25.286 opposite 4. 

Fifth column: 20, 19.5, 21.3, 23, 22.3, 21.431, 20.4, 21.8, 24.1. 













r'" 




il 

Datum Level. 






V. 


1 


2 3 4 


5 


6 



3. Column a^.G^. 20.8, 20.4, 20.0, 19.6, etc. 
Column C. 0.0, 5.3, 6.4, 7.4, 5.0, 5.1, etc. 



cq lo o o c<« O 




8 9 10 10.25 



70 




TABLE VII.- 


-TRAVERSE TABLE. 






Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 


o f 


Lat. 


Dep, 


Lat. Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


015 


1.000 


0.004 


2.000 0.009 


3.000 


0.013 


4.000 


0.017 


5.000 


0.022 


89 45 


30 


1.000 


0.009 


2.000 0.017 


3.000 


0.026 


4.000 


0.035 


5.000 


0.044 


30 


45 


1.000 


0.013 


2.000 0.026 


3.000 


0.039 


4.000 


0.052 


5.000 


0.065 


15 


1 


1.000 


0.017 


2.000 0.035 


3.000 


0.052 


3.999 


0.070 


4.999 


0.087 


89 


15 


1.000 


0.022 


2.000 0.044 


2.999 


0.065 


3.999 


0.087 


4.999 


0.109 


45 


30 


1.000 


0.026 


1.999 0.052 


2.999 


0.079 


3.999 


0.105 


4.998 


0.131 


30 


45 


1.000 


0.031 


1.999 0.061 


2.999 


0.092 


3.998 


0.122 


4.998 


0.153 


15 


2 


0.999 


0.035 


1.999 0.070 


2.998 


0.105 


3.998 


0.140 


4.997 


0.174 


88 


15 


0.999 


0.039 


1.998 0.079 


2.998 


0.118 


3.997 


0.157 


4.996 


0.196 


45 


30 


0.999 


0.044 


1.998 0.087 


2.997 


0.131 


3.996 


0.174 


4.995 


0.218 


30 


45 


0.999 


0.048 


1.998 '0.096 


2.997 


0.144 


3.995 


0.192 


4.994 


0.240 


15 


3 


0.999 


0.052 


1.997 0.105 


2.996 


0.157 


3.995 


0.209 


4.993 


0.262 


87 


15 


0.998 


0.057 


1.997 0.113 


2.995 


0.170 


3.994 


0.227 


4.992 


0.283 


45 


30 


0.998 


0.061 


1.996 0.122 


2.994 


0.183 


3.993 


0.244 


4.991 


0.305 


30 


45 


0.998 


0.065 


1.996 0.131 


2.994 


0.196 


3.991 


0.262 


4.989 


0.327 


15 


4 


0.998 


0.070 


1.995 0.140 


2.993 


0.209 


3.990 


0.279 


4.988 


0,349 


86 


15 


0.997 


0.074 


1.995 0.148 


2.992 


0.222 


3.989 


0.296 


4.986 


0.371 


45 


30 


0.997 


0.078 


1.994 0.157 


2.991 


0.235 


3.988 


0.314 


4.985 


0.392 


30 


45 


0.997 


0.083 


1.993 0.166 


2.990 


0.248 


3.986 


0.331 


4.983 


0.414 


15 


6 


0.996 


0.087 


1.992 0.174 


2.989 


.0.261 


3.985 


0.349 


4.981 


0.436 


85 


15 


0.996 


0.092 


1.992 0.183 


2.987 


0.275 


3.983 


0.366 


4.979 


0.458 


45 


30 


0.995 


0.096 


1.991 0.192 


2.986 


0.288 


3.982 


0.383 


4.977 


0.479 


30 


45 


0.995 


0.100 


1.990 0.200 


2.985 


0.301 


3.980 


0.401 


4.975 


0.501 


15 


6 


0.995 


0.105 


1.989 0.209 


2.984 


0.314 


3.978 


0.418 


4.973 


0.523 


84 


15 


0.994 


0.109 


1.988 0.218 


2.982 


0.327 


3.976 


0.435 


4.970 


0.544 


45 


30 


0.994 


0.113 


1.987 0.226 


2.981 


0.340 


3.974 


0.453 


4.968 


0.566 


30 


45 


0.993 


0.118 


1.986 0.235 


2.979 


0.353 


3.972 


0.470 


4.965 


0.588 


15 


7 


0.993 


0.122 


1.985 0.244 


2.978 


0.366 


3.970 


0.487 


4.963 


0.609 


83 


15 


0.992 


0.126 


1.984 0.252 


2.976 


0.379 


3.968 


0.505 


4.960 


0.631 


45 


30 


0.991 


0.131 


1.983 0.261 


2.974 


0.392 


3.966 


0.522 


4.957 


0.653 


30 


45 


0.991 


0.135 


1.982 0.270 


2.973 


0.405 


3.963 


0.539 


4.954 


0.674 


15 


8 


0.990 


0.139 


1.981 0.278 


2.971 


0.418 


3.961 


0.557 


4.951 


0.696 


82 


15 


0.990 


0.143 


1.979 0.287 


2.969 


0.430 


3.959 


0.574 


4.948 


0.717 


45 


30 


0.989 


0.148 


1.978 0.296 


2.967 


0.443 


3.956 


0.591 


4.945 


0.739 


30 


45 


0.988 


0.152 


1.977 0.304 


2.965 


0.456 


3.953 


0.608 


4.942 


0.761 


15 


9 


0.988 


0.156 


1.975 0.313 


2.963 


0.469 


3.951 


0.626 


4.938 


0.782 


81 


15 


0.987 


0.161 


1.974 0.321 


2.961 


0.482 


3.948 


0.643 


4.935 


0.804 


45 


30 


0.986 


0.165 


1.973 0.330 


2.959 


0.495 


3.945 


0.660 


4.931 


0.825 


30 


45 


0.986 


0.169 


1.971 0.339 


2.957 


0.508 


3.942 


0.677 


4.928 


0.847 


15 


10 


0.985 


0.174 


1.970 0.347 


2.954 


0.521 


3.939 


0.695 


4.924 


0.868 


80 


15 


0.984 


0.178 


1.968 0.356 


2.952 


0.534 


3.936 


0.712 


4.920 


0.890 


45 


30 


0.983 


0.182 


1.967 0.364 


2.950 


0.547 


3.933 


0.729 


4.916 


0.911 


30 


45 


0.982 


0.187 


1.965 0.373 


2.947 


0.560 


3.930 


0.746 


4.912 


0.933 


15 


11 


0.982 


0.191 


1.963 0.382 


2.945 


0.572 


3.927 


0.763 


4.908 


0.954 


79 


15 


0.981 


0.195 


1.962 0.390 


2.942 


0.585 


3.923 


0.780 


4.904 


0.975 


45 


30 


0.980 


0.199 


1.960 0.399 


2.940 


0.598 


3.920 


0.797 


4.900 


0.997 


30 


45 


0.979 


0.204 


1.958 0.407 


2.937 


0611 


3.916 


0.815 


4.895 


1.018 


15 


12 


0.978 


0.208 


1.956 0.416 


2.934 


0.624 


3.913 


0.832 


4.891 


1.040 


78 


15 


0.977 


0.212 


1.954 0.424 


2.932 


0.637 


3.909 


0.849 


4.886 


1.061 


45 


30 


0.976 


0.216 


1.953 0.433 


2.929 


0.649 


3.905 


0.866 


4.881 


1.082 


30 


45 


0.975 


0.221 


1.951 0.441 


2.926 


0.662 


3.901 


0.883 


4.877 


1.103 


15 


13 


0.974 


0.225 


1.949 0.450 


2.923 


0.675 


3.897 


0.900 


4.872 


1.125 


77 


15 


0.973 


0.229 


1.947 0.458 


2.920 


0.688 


3.894 


0.917 


4.867 


1.146 


45 


30 


0.972 


0.233 


1.945 0.467 


2.917 


0.700 


3.889 


0.934 


4.862 


1.167 


30 


45 


0.971 


0.238 


1.943 0.475 


2.914 


0.713 


3.885 


0.951 


4.857 


1.188 


15 


14 


0.970 


0.242 


1.941 0.484 


2.911 


0.726 


3.881 


0.968 


4.851 


1.210 


76 


15 


0.969 


0.246 


1.938 0.492 


2.908 


0.738 


3.877 


0.985 


4.846 


1.231 


45 


30 


0.968 


0.250 


1.936 0.501 


2.904 


0.751 


3.873 


1.002 


4.841 


1.252 


30 


45 


0.967 


0.255 


1.934 0.509 


2.901 


0.764 


3.868 


1.018 


4.835 


1.273 


15 


16 


0.966 


0.259 


1.932 0.518 


2.898 


0.776 


3.864 


1.035 


4.830 


1.294 


75 


o r 


Dap. Lat. 
Distance 1. 


Dep. Lat. 
Distance 2. 


Dep. Lat. 
Distance 3. 


Dep. Lat. 
Distance 4. 


Dep. Lat. 
Distance 5. 


O f 


Bearing. 


Bearing. 



76°-90^ 













o°- 


-16° 










71 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 


o f 


Lat. 


Dep, 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


15 


6.000 


0.026 


7.000 


0.031 


8.000 


0.035 


9.000 


0.039 


10.000 


0.044 


89 45 


30 


6.000 


0.052 


7.000 


0.061 


8.000 


0.070 


9.000 


0.079 


10.000 


0.087 


30 


45 


5.999 


0.079 


6.999 


0.092 


7.999 


0.105 


8.999 


0.118 


9.999 


0.131 


15 


1 


5.999 


0.105 


6.999 


0.122 


7.999 


0.140 


8.999 


0.157 


9.999 


0.175 


89 


15 


5.999 


0.131 


6.998 


0.153 


7.998 


0.175 


8.998 


0.196 


9.998 


0.218 


45 


30 


5.998 


0.157 


6.998 


0.183 


7.997 


0.209 


8.997 


0.236 


9.997 


0.262 


30 


45 


5.997 


0.183 


6.997 


0.214 


7.996 


0.244 


8.996 


0.275 


9.995 


0.305 


15 


2 


5.996 


0.209 


6.996 


0.244 


7.995 


0.279 


8.995 


0.314 


9.994 


0.349 


88 


15 


5.995 


0.236 


6.995 


0.275 


7.994 


0.314 


8.993 


0.353 


9.992 


0.393 


45 


30 


5.994 


0.262 


6.993 


0.305 


7.992 


0.349 


8.991 


0.393 


9.991 


0.436 


30 


45 


5.993 


0.288 


6.992 


0.336 


7.991 


0.384 


8.990 


0.432 


9.989 


0.480 


15 


3 


5.992 


0.314 


6.990 


0.366 


7.989 


0.419 


8.988 


0.471 


9.986 


0.523 


87 


15 


5.990 


0.340 


6.989 


0.397 


7.987 


0.454 


8.986 


0.510 


9.984 


0.567 


45 


30 


5.989 


0.366 


6.987 


0.427 


7.985 


0.488 


8.983 


0.549 


9.981 


0.611 


30 


45 


5.987 


0.392 


6.985 


0.458 


7.983 


0.523 


8.981 


0.589 


9.979 


0.654 


15 


4 


5.985 


0.419 


6.983 


0.488 


7.981 


0.558 


8.978 


0.628 


9.976 


0.698 


86 


15 


5.984 


0.445 


6.981 


0.519 


7.978 


0.593 


8.975 


0.667 


9.973 


0.741 


45 


30 


5.982 


0.471 


6.978 


0.549 


7.975 


0.628 


8.972 


0.706 


9.969 


0.785 


30 


45 


5.979 


0.497 


6.976 


0.580 


7.973 


0.662 


8.969 


0.745 


9.966 


0.828 


15 


5 


5.977 


0.523 


6.973 


0.610 


7.970 


0.697 


8.966 


0.784 


9.962 


0.872 


85 


15 


5.975 


0.549 


6.971 


0.641 


7.966 


0.732 


8.962 


0.824 


9.958 


0.915 


45 


30 


5.972 


0.575 


6.968 


0.671 


7.963 


0.767 


8.959 


0.863 


9.954 


0.959 


30 


45 


5.970 


0.601 


6.965 


0.701 


7.960 


0.802 


8.955 


0.902 


9.950 


1.002 


15 


6 


5.967 


0.627 


6.962 


0.732 


7.956 


0.836 


8.951 


0.941 


9.945 


1.045 


84 


15 


5.964 


0.653 


6.958 


0.762 


7.952 


0.871 


8.947 


0.980 


9.941 


1.089 


45 


30 


5.961 


0.679 


6.955 


0.792 


7.949 


0.906 


8.942 


1.019 


9.936 


1.132 


30 


45 


5.958 


0.705 


6.951 


0.823 


7.945 


0.940 


8.938 


1.058 


9.931 


1.175 


15 


7 


5.955 


0.731 


6.948 


0.853 


7.940 


0.975 


8.933 


1.097 


9.926 


1.219 


83 


15 


5.952 


0.757 


6.944 


0.883 


7.936 


1.010 


8.928 


1.136 


9.920 


1.262 


45 


30 


5.949 


0.783 


6.940 


0.914 


7.932 


1.044 


8.923 


1.175 


9.914 


1.305 


30 


45 


5.945 


0.809 


6.936 


0.944 


7.927 


1.079 


8.918 


1.214 


9.909 


1.349 


15 


8 


5.942 


0.835 


6.932 


0.974 


7.922 


1.113 


8.912 


1.253 


9.903 


1.392 


82 


15 


5.938 


0.861 


6.928 


1.004 


7.917 


1.148 


8.907 


1.291 


9.897 


1.435 


45 


30 


5.934 


0.887 


6.923 


1.035 


7.912 


1.182 


8.901 


1.330 


9.890 


1.478 


30 


45 


5.930 


0.913 


6.919 


1.065 


7.907 


1.217 


8.895 


1.369 


9.884 


1.521 


15 


9 


5.926 


0.939 


6.914 


1.095 


7.902 


1.251 


8.889 


1.408 


9.877 


1.564 


81 


15 


5.922 


0.964 


6.909 


1.125 


7.896 


1.286 


8.883 


1.447 


9.870 


1.607 


45 


30 


5.918 


0.990 


6.904 


1.155 


7.890 


1.320 


8.877 


1.485 


9.863 


1.651 


30 


45 


5.913 


1.016 


6.899 


1.185 


7.884 


1.355 


8.870 


1.524 


9.856 


1.694 


15 


10 


5.909 


1.042 


6.894 


1.216 


7.878 


1.389 


8.863 


1.563 


9.848 


1.737 


80 


15 


5.904 


1.068 


6.888 


1.246 


7.872 


1.424 


8.856 


1.601 


9.840 


1.779 


45 


30 


5.900 


1.093 


6.883 


1.276 


7.866 


1.458 


8.849 


1.640 


9.833 


1.822 


30 


45 


5.895 


1.119 


6.877 


1.306 


7.860 


1.492 


8.842 


1.679 


9.825 


1.865 


15 


11 


5.890 


1.145 


6.871 


1.336 


7.853 


1.526 


8.835 


1.717 


9.816 


1.908 


79 


15 


5.885 


1.171 


6.866 


1.366 


7.846 


1.561 


8.827 


1.756 


9.808 


1.951 


45 


30 


5.880 


1.196 


6.859 


1.396 


7.839 


1.595 


8.819 


1.794 


9.799 


1.994 


30 


45 


5.874 


1.222 


6.853 


1.425 


7.832 


1.629 


8.811 


1.833 


9.791 


2.036 


15 


12 


5.869 


1.247 


6.847 


1.455 


7.825 


1.663 


8.803 


1.871 


9.782 


2.079 


78 


15 


5.863 


1.273 


6.841 


1.485 


7.818 


1.697 


8.795 


1.910 


9.772 


2.122 


45 


30 


5.858 


1.299 


6.834 


1.515 


7.810 


1.732 


8.787 


1.948 


9.763 


2.164 


30 


45 


5.852 


1.324 


6.827 


1.545 


7.803 


1.766 


8.778 


1.986 


9.753 


2.207 


15 


13 


5.846 


1.350 


6.821 


1.575 


7.795 


1.800 


8.769 


2.025 


9.744 


2.250 


77 


15 


5.840 


1.375 


6.814 


1.604 


7.787 


1.834 


8.760 


2.063 


9.734 


2.292 


45 


30 


5.834 


1.401 


6.807 


1.634 


7.779 


1.868 


8.751 


2.101 


9.724 


2.335 


30 


45 


5.828 


1.426 


6.799 


1.664 


7.771 


1.902 


8.742 


2.139 


9.713 


2.377 


15 


14 


5.822 


1.452 


6.792 


1.693 


7.762 


1.935 


8.733 


2.177 


9.703 


2.419 


76 


15 


5.815 


1.477 


6.785 


1.723 


7.754 


1.969 


8.723 


2.215 


9.692 


2.462 


45 


30 


5.809 


1.502 


6.777 


1.753 


7.745 


2.003 


8.713 


2.253 


9.682 


2.504 


30 


45 


5.802 


1.528 


6.769 


1.782 


7.736 


2.037 


8.703 


2.291 


9.671 


2.546 


15 


15 


5.796 


1.553 


6.761 


1.812 


7.727 


2.071 


8.693 


2.329 


9.659 


2.588 


75 


o r 


Dep. Lat. 
Distance 6. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


o f 


Bearing, 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 



76° -90' 



72 










15°- 


-30° 










Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 


o f 


Lat. 


Dep, 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


15 15 


0.965 


0.263 


1.930 


0.526 


2.894 


0.789 


3.859 


1.052 


4.824 


1.315 


74 45 


30 


0.964 


0.267 


1.927 


0.534 


2.891 


0.802 


3.855 


1.069 


4.818 


1.336 


30 


45 


0.962 


0.271 


1.925 


0.543 


2.887 


0.814 


3.850 


1.086 


4.812 


1.357 


15 


16 


0.961 


0.276 


1.923 


0.551 


2.884 


0.827 


3.845 


1.103 


4.806 


1.378 


74 


15 


0.960 


0.280 


1.920 


0.560 


2.880 


0.839 


3.840 


1.119 


4.800 


1.399 


45 


30 


0.959 


0.284 


1.918 


0.568 


2.876 


0.852 


3.835 


1.136 


4.794 


1.420 


30 


45 


0.958 


0.288 


1.915 


0.576 


2.873 


0.865 


3.830 


1.153 


4.788 


1.441 


15 


17 


0.956 


0.292 


1.913 


0.585 


2.869 


0.877 


3.825 


1.169 


4.782 


1.462 


73 


15 


0.955 


0.297 


1.910 


0.593 


2.865 


0.890 


3.820 


1.186 


4.775 


1.483 


45 


30 


0.954 


0.301 


1.907 


0.601 


2.861 


0.902 


3.815 


1.203 


4.769 


1.504 


30 


45 


0.952 


0.305 


1.905 


0.610 


2.857 


0.915 


3.810 


1.220 


4.762 


1.524 


15 


18 


0.951 


0.309 


1.902 


0.618 


2.853 


0.927 


3.804 


1.236 


4.755 


1.545 


72 


15 


0.950 


0.313 


1.899 


0.626 


2.849 


0.939 


3.799 


1.253 


4.748 


1.566 


45 


30 


0.948 


0.317 


1.897 


0.635 


2.845 


0.952 


3.793 


1.269 


4.742 


1.587 


30 


45 


0.947 


0.321 


1.894 


0.643 


2.841 


0.964 


3.788 


1.286 


4.735 


1.607 


15 


19 


0.946 


0.326 


1.891 


0.651 


2.837 


0.977 


3.782 


1.302 


4.728 


1.628 


71 


15 


0.944 


0.330 


1.888 


0.659 


2.832 


0.989 


3.776 


1.319 


4.720 


1.648 


45 


30 


0.943 


0.334 


1.885 


0.668 


2.828 


1.001 


3.771 


1.335 


4.713 


1.669 


30 


45 


0.941 


0.338 


1.882 


0.676 


2.824 


1.014 


3.765 


1.352 


4.706 


1.690 


15 


20 


0.940 


0.342 


1.879 


0.684 


2.819 


1.026 


3.759 


1.368 


4.698 


1.710 


70 


IS 


0.938 


0.346 


1.876 


0.692 


2.815 


1.038 


3.753 


1.384 


4.691 


1.731 


45 


30 


0.937 


0.350 


1.873 


0.700 


2.810 


1.051 


3.747 


1.401 


4.683 


1.751 


30 


45 


0.935 


0.354 


1.870 


0.709 


2.805 


1.063 


3.741 


1.417 


4.676 


1.771 


15 


21 


0.934 


0.358 


1.867 


0.717 


2.801 


1.075 


3.734 


1.433 


4.668 


1.792 


69 


15 


0.932 


0.362 


1.864 


0.725 


2.796 


1.087 


3.728 


1.450 


4.660 


1.812 


45 


30 


0.930 


0.367 


1.861 


0.733 


2.791 


1.100 


3.722 


1.466 


4.652 


1.833 


30 


45 


0.929 


0.371 


1.858 


0.741 


2.786 


1.112 


3.715 


1.482 


4.644 


1.853 


15 


22 


0.927 


0.375 


1.854 


0.749 


2.782 


1.124 


3.709 


1.498 


4.636 


1.873 


68 


15 


0.926 


0.379 


1.851 


0.757 


2.777 


1.136 


3.702 


1.515 


4.628 


1.893 


45 


30 


0.924 


0.383 


1.848 


0.765 


2.772 


1.148 


3.696 


1.531 


4.619 


1.913 


30 


45 


0.922 


0.387 


1.844 


0.773 


2.767 


1.160 


3.689 


1.547 


4.611 


1.934 


15 


23 


0.921 


0.391 


1.841 


0.781 


2.762 


1.172 


3.682 


1.563 


4.603 


1.954 


67 


15 


0.919 


0.395 


1.838 


0.789 


2.756 


1.184 


3.675 


1.579 


4.594 


1.974 


45 


30 


0.917 


0.399 


1.834 


0.797 


2.751 


1.196 


3.668 


1.595 


4.585 


1.994 


30 


45 


0.915 


0.403 


1.831 


0.805 


2.746 


1.208 


3.661 


1.611 


4.577 


2.014 


15 


24 


0.914 


0.407 


1.827 


0.813 


2.741 


1.220 


3.654 


1.627 


4.568 


2.034 


66 


15 


0.912 


0.411 


1.824 


0.821 


2.735 


1.232 


3.647 


1.643 


4.559 


2.054 


45 


30 


0.910 


0.415 


1.820 


0.829 


2.730 


1.244 


3.640 


1.659 


4.550 


2.073 


30 


45 


0.908 


0.419 


1.816 


0.837 


2.724 


1.256 


3.633 


1.675 


4.541 


2.093 


15 


25 


0.906 


0.423 


1.813 


0.845 


2.719 


1.268 


3.625 


1.690 


4.532 


2.113 


65 


15 


0.904 


0.427 


1.809 


0.853 


2.713 


1.280 


3.618 


1.706 


4.522 


2.133 


45 


30 


0.903 


0.431 


1.805 


0.861 


2.708 


1.292 


3.610 


1.722 


4.513 


2.153 


30 


45 


0.901 


0.434 


1.801 


0.869 


2.702 


1.303 


3.603 


1.738 


4.503 


2.172 


15 


26 


0.899 


0.438 


1.798 


0.877 


2.696 


1.315 


3.595 


1.753 


4.494 


2.192 


64 


15 


897 


0.442 


1.794 


0.885 


2.691 


1.327 


3.587 


1.769 


4.484 


2.211 


45 


30 


0.895 


0.446 


1.790 


892 


2.685 


1.339 


3.580 


1.785 


4.475 


2.231 


30 


45 


0.893 


0.450 


1.786 


0.900 


2.679 


1.350 


3.572 


1.800 


4.465 


2.250 


15 


27 


0.891 


0.454 


1.782 


0.908 


2.673 


1.362 


3.564 


1.816 


4.455 


2.270 


63 


15 


0.889 


0.458 


1.778 


0.916 


2.667 


1.374 


3.556 


1.831 


4.445 


2.289 


45 


30 


0.887 


0.462 


1.774 


0.923 


2.661 


1.385 


3.548 


1.847 


4.435 


2.309 


30 


45 


0.885 


0.466 


1.770 


0.931 


2.655 


1.397 


3.540 


1.862 


4.425 


2.328 


15 


28 


0.883 


0.469 


1.766 


0.939 


2.649 


1.408 


3.532 


1.878 


4.415 


2.347 


62 


15 


0.881 


0.473 


1.762 


0.947 


2.643 


1.420 


3.524 


1.893 


4.404 


2.367 


45 


30 


0.879 


0.477 


1.758 


0.954 


2.636 


1.431 


3.515 


1.909 


4.394 


2.386 


30 


45 


0.877 


0.481 


1.753 


0.962 


2.630 


1.443 


3.507 


1.924 


4.384 


2.405 


15 


29 


0.875 


0.485 


1.749 


0.970 


2.624 


1.454 


3.498 


1.939 


4.373 


2.424 


61 


15 


0.872 


0.489 


1.745 


0.977 


2.617 


1.466 


3.490 


1.954 


4.362 


2.443 


45 


30 


0.870 


0.492 


1.741 


0.985 


2.611 


1.477 


3.481 


1.970 


4.352 


2.462 


30 


45 


0.868 


0.496 


1.736 


0.992 


2.605 


1.489 


3.473 


1.985 


4.341 


2.481 


15 


30 


0.866 


0.500 


1.732 


1.000 


2.598 


1.500 


3.464 


2.000 


4.330 


2.500 


60 


O f 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


O f 


Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 



60° - 75^ 













16°- 


-30 


O 








73 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 


o r 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


15 15 


5.789 


1.578 


6.754 


1.841 


7.718 


2.104 


8.683 


2.367 


9.648 


2.630 


74 45 


30 


5.782 


1.603 


6.745 


1.871 


7.709 


2.138 


8.673 


2.405 


9.636 


2.672 


30 


45 


5.775 


1.629 


6.737 


1.900 


7.700 


2.172 


8.662 


2.443 


9.625 


2.714 


15 


16 


5.768 


1.654 


6.729 


1.929 


7.690 


2.205 


8.651 


2.481 


9.613 


2.756 


74 


15 


5.760 


1.679 


6.720 


1.959 


7.680 


2.239 


8.640 


2.518 


9.601 


2.798 


45 


30 


5.753 


1.704 


6.712 


1.988 


7.671 


2.272 


8.629 


2.556 


9.588 


2.840 


30 


45 


5.745 


1.729 


6.703 


2.017 


7.661 


2.306 


8.618 


2.594 


9.576 


2.882 


15 


17 


5.738 


1.754 


6.694 


2.047 


7.650 


2.339 


8.607 


2.631 


9.563 


2.924 


73 


15 


5.730 


1.779 


6.685 


2.076 


7.640 


2.372 


8.595 


2.669 


9.550 


2.965 


45 


30 


5.722 


1.804 


6.676 


2.105 


7.630 


2.406 


8.583 


2.706 


9.537 


3.007 


30 


45 


5.714 


1.829 


6.667 


2.134 


7.619 


2.439 


8.572 


2.744 


9.524 


3.049 


15 


18 


5.706 


1.854 


6.657 


2.163 


7.608 


2.472 


8.560 


2.781 


9.511 


3.090 


72 


15 


5.698 


1.879 


6.648 


2.192 


7.598 


2.505 


8.547 


2.818 


9.497 


3.132 


45 


30 


5.690 


1.904 


6.638 


2.221 


7.587 


2.538 


8.535 


2.856 


9.483 


3.173 


30 


45 


5.682 


1.929 


6.629 


2.250 


7.575 


2.572 


8.522 


2.893 


9.469 


3.214 


15 


19 


5.673 


1.953 


6.619 


2.279 


7.564 


2.605 


8.510 


2.930 


9.455 


3.256 


71 


15 


5.665 


1.978 


6.609 


2.308 


7.553 


2.638 


8.497 


2.967 


9.441 


3.297 


45 


30 


5.656 


2.003 


6.598 


2.337 


7.541 


2.670 


8.484 


3.004 


9.426 


3.338 


30 


45 


5.647 


2.028 


6.588 


2.365 


7.529 


2.703 


8.471 


3.041 


9.412 


3.379 


15 


20 


5.638 


2.052 


6.578 


2.394 


7.518 


2.736 


8.457 


3.078 


9.397 


3.420 


70 


15 


5.629 


2.077 


6.567 


2.423 


7.506 


2.769 


8.444 


3.115 


9.382 


3.461 


45 


30 


5.620 


2.101 


6.557 


2.451 


7.493 


2.802 


8.430 


3.152 


9.367 


3.502 


30 


45 


5.611 


2.126 


6.546 


2.480 


7.481 


2.834 


8.416 


3.189 


9.351 


3.543 


15 


21 


5.601 


2.150 


6.535 


2.509 


7.469 


2.867 


8.402 


3.225 


9.336 


3.584 


69 


15 


5.592 


2.175 


6.524 


2.537 


7.456 


2.900 


8.388 


3.262 


9.320 


3.624 


45 


30 


5.582 


2.199 


6.513 


2.566 


7.443 


2.932 


8.374 


3.299 


9.304 


3.665 


30 


45 


5.573 


2.223 


6.502 


2.594 


7.430 


2.964 


8.359 


3.335 


9.288 


3.706 


15 


22 


5.563 


2.248 


6.490 


2.622 


7.417 


2.997 


8.345 


3.371 


9.272 


3.746 


68 


15 


5.553 


2.272 


6.479 


2.651 


7.404 


3.029 


8.330 


3.408 


9.255 


3.787 


45 


30 


5.543 


2.296 


6.467 


2.679 


7.391 


3.061 


8.315 


3.444 


9.239 


3.827 


30 


45 


5.533 


2.320 


6.455 


2.707 


7.378 


3.094 


8.300 


3.480 


9.222 


3.867 


15 


23 


5.523 


2.344 


6.444 


2.735 


7.364 


3.126 


8.285 


3.517 


9.205 


3.907 


67 


15 


5.513 


2.368 


6.432 


2.763 


7.350 


3.158 


8.269 


3.553 


9.188 


3.947 


45 


30 


5.502 


2.392 


6.419 


2.791 


7.336 


3.190 


8.254 


3.589 


9.171 


3.988 


30 


45 


5.492 


2.416 


6.407 


2.819 


7.322 


3.222 


8.238 


3.625 


9.153 


4.028 


15 


24 


5.481 


2.440 


6.395 


2.847 


7.308 


3.254 


8.222 


3.661 


9.136 


4.067 


66 


15 


5.471 


2.464 


6.382 


2.875 


7.294 


3.286 


8.206 


3.696 


9.118 


4.107 


45 


30 


5.460 


2.488 


6.370 


2.903 


7.280 


3.318 


8.190 


3.732 


9.100 


4.147 


30 


45 


5.449 


2.512 


6.357 


2.931 


7.265 


3.349 


8.173 


3.768 


9.081 


4.187 


15 


25 


5.438 


2.536 


6.344 


2.958 


7.250 


3.381 


8.157 


3.804 


9.063 


4.226 


65 


15 


5.427 


2.559 


6.331 


2.986 


7.236 


3.413 


8.140 


3.839 


9.045 


4.266 


45 


30 


5.416 


2.583 


6.318 


3.014 


7.221 


3.444 


8.123 


3.875 


9.026 


4.305 


30 


45 


5.404 


2.607 


6.305 


3.041 


7.206 


3.476 


8.106 


3.910 


9.007 


4.345 


15 


26 


5.393 


2.630 


6.292 


3.069 


7.190 


3.507 


8.089 


3.945 


8.988 


4.384 


64 


15 


5.381 


2.654 


6.278 


3.096 


7.175 


3.538 


8.072 


3.981 


8.969 


4.423 


45 


30 


5.370 


2.677 


6.265 


3.123 


7.160 


3.570 


8.054 


4.016 


8.949 


4.462 


30 


45 


5.358 


2.701 


6.251 


3.151 


7.144 


3.601 


8.037 


4.051 


8.930 


4.501 


15 


2T 


5.346 


2.724 


6.237 


3.178 


7.128 


3.632 


8.019 


4.086 


8.910 


4.540 


63 


15 


5.334 


2.747 


6.223 


3.205 


7.112 


3.663 


8.001 


4.121 


8.890 


4.579 


45 


30 


5.322 


2.770 


6.209 


3.232 


7.096 


3.694 


7.983 


4.156 


8.870 


4.618 


30 


45 


5.310 


2.794 


6.195 


3.259 


7.080 


3.725 


7.965 


4.190 


8.850 


4.656 


15 


28 


5.298 


2.817 


6.181 


3.286 


7.064 


3.756 


7.947 


4.225 


. 8.829 


4.695 


62 


15 


5.285 


2.840 


6.166 


3.313 


7.047 


3.787 


7.928 


4.260 


8.809 


4.733 


45 


30 


5.273 


2.863 


6.152 


3.340 


7.031 


3.817 


7.909 


4.294 


8.788 


4.772 


30 


45 


5.260 


2.886 


6.137 


3.367 


7.014 


3.848 


7.891 


4329 


8.767 


4.810 


15 


29 


5.248 


2.909 


6.122 


3.394 


6.997 


3.878 


7.872 


4.363 


8.746 


4.848 


61 


15 


5.235 


2.932 


6.107 


3.420 


6.980 


3.909 


7.852 


4.398 


8.725 


4.886 


45 


30 


5.222 


2.955 


6.093 


3.447 


6.963 


3.939 


7.833 


4.432 


8.704 


4.924 


30 


45 


5.209 


2.977 


6.077 


3.474 


6.946 


3.970 


7.814 


4.466 


8.682 


4.962 


15 


30 


5.196 


3.000 


6.062 


3.500 


6.928 


4.000 


7.794 


4.500 


8.660 


5.000 


60 


o r 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


O f 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 



60° -76' 



74 










30° -45 


o 










Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 


O f 


Lat. 


Dep. 


Lat. 


Dep. 


Lat, Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


30 15 


0.864 


0.504 


1.728 


1.008 


2.592 1.511 


3.455 


2.015 


4.319 


2.519 


59 45 


30 


0.862 


0.508 


1.723 


1.015 


2.585 1.523 


3.447 


2.030 


4.308 


2.538 


30 


45 


0.859 


0.511 


1.719 


1.023 


2.578 1.534 


3.438 


2.045 


4.297 


2.556 


15 


31 


0.857 


0.515 


1.714 


1.030 


2.572 1.545 


3.429 


2.060 


4.286 


2.575 


59 


15 


0.855 


0.519 


1.710 


1.038 


2.565 1.556 


3.420 


2.075 


4.275 


2.594 


45 


30 


0.853 


0.522 


1.705 


1.045 


2.558 1.567 


3.411 


2.090 


4.263 


2.612 


30 


45 


0.850 


0.526 


1.701 


1.052 


2.551 1.579 


3.401 


2.105 


4.252 


2.631 


15 


32 


0.848 


0.530 


1.696 


1.060 


2.544 1.590 


3.392 


2.120 


4.240 


2.650 


58 


15 


0.846 


0.534 


1.691 


1.067 


2.537 1.601 


3.383 


2.134 


4.229 


2.668 


45 


30 


0.843 


0.537 


1.687 


1.075 


2.530 1.612 


3.374 


2.149 


4.217 


2.686 


30 


45 


0.841 


0.541 


1.682 


1.082 


2.523 1.623 


3.364 


2.164 


4.205 


2.705 


15 


33 


0.839 


0.545 


1.677 


1.089 


2.516 1.634 


3.355 


2.179 


4.193 


2.723 


57 


IS 


0.836 


0.548 


1.673 


1.097 


2.509 1.645 


3.345 


2.193 


4.181 


2.741 


45 


30 


0.834 


0.552 


1.668 


1.104 


2.502 1.656 


3.336 


2.208 


4.169 


2.760 


30 


45 


0.831 


0.556 


1.663 


1.111 


2.494 1.667 


3.326 


2.222 


4.157 


2.778 


15 


34 


0.829 


0.559 


1.658 


1.118 


2.487 1.678 


3.316 


2.237 


4.145 


2.796 


56 


15 


0.827 


0.563 


1.653 


1.126 


2.480 1.688 


3.306 


2.251 


4.133 


2.814 


45 


30 


0.824 


0.566 


1.648 


1.133 


2.472 1.699 


3.297 


2.266 


4.121 


2.832 


30 


45 


0.822 


0.570 


1.643 


1.140 


2.465 1.710 


3.287 


2.280 


4.108 


2.850 


15 


35 


0.819 


0.574 


1.638 


1.147 


2.457 1.721 


3.277 


2.294 


4.096 


2.868 


^^ 


15 


0.817 


0.577 


1.633 


1.154 


2.450' 1.731 


3.267 


2.309 


4.083 


2.886 


45 


30 


0.814 


0.581 


1.628 


1.161 


2.442 1.742 


3.257 


2.323 


4.071 


2.904 


30 


45 


0.812 


0.584 


1.623 


1.168 


2.435 1.753 


3.246 


2.337 


4.058 


2.921 


IS 


36 


0.809 


0.588 


1.618 


1.176 


2.427 1.763 


3.236 


2.351 


4.045 


2.939 


54 


15 


0.806 


0.591 


1.613 


1.183 


2.419 1.774 


3.226 


2.365 


4.032 


2.957 


45 


30 


0.804 


0.595 


1.608 


1.190 


2.412 1.784 


3.215 


2.379 


4.019 


2.974 


30 


45 


0.801 


0.598 


1.603 


1.197 


2.404 1.795 


3.205 


2.393 


4.006 


2.992 


15 


37 


0.799 


0.602 


1.597 


1.204 


2.396 1.805 


3.195 


2.407 


3.993 


3.009 


53 


15 


0.796 


0.605 


1.592 


1.211 


2.388 1.816 


3.184 


2.421 


3.980 


3.026 


45 


30 


0.793 


0.609 


1.587 


1.218 


2.380 1.826 


3.173 


2.435 


3.967 


3.044 


30 


45 


0.791 


0.612 


1.581 


1.224 


2.372 1.837 


3.163 


2.449 


3.953 


3.061 


15 


38 


0.788 


0.616 


1.576 


1.231 


2.364 1.847 


3.152 


2.463 


3.940 


3.078 


52 


15 


0.785 


0.619 


1.571 


1.238 


2.356 1.857 


3.141 


2.476 


3.927 


3.095 


45 


30 


0.783 


0.623 


1.565 


1.245 


2.348 1.868 


3.130 


2.490 


3.913 


3.113 


30 


45 


0.780 


0.626 


1.560 


1.252 


2.340 1.878 


3.120 


2.504 


3.899 


3.130 


15 


39 


0.777 


0.629 


1.554 


1.259 


2.331 1.888 


3.109 


2.517 


3.886 


3.147 


51 


15 


0.774 


0.633 


1.549 


1.265 


2.323 1.898 


3.098 


2.531 


3.872 


3.164 


45 


30 


0.772 


0.636 


1.543 


1.272 


2.315 1.908 


3.086 


2.544 


3.858 


3.180 


30 


45 


0.769 


0.639 


1.538 


1.279 


2.307 1.918 


3.075 


2.558 


3.844 


3.197 


15 


40 


0.766 


0.643 


1.532 


1.286 


2.298 1.928 


3.064 


2.571 


3.830 


3.214 


60 


15 


0.763 


0.646 


1.526 


1.292 


2.290 1.938 


3.053 


2.584 


3.816 


3.231 


45 


30 


0.760 


0.649 


1.521 


1.299 


2.281 1.948 


3.042 


2.598 


3.802 


3.247 


30 


45 


0.758 


0.653 


1.515 


1.306 


2.273 1.958 


3.030 


2.611 


3.788 


3.264 


15 


41 


0.755 


0.656 


1.509 


1.312 


2.264 1.968 


3.019 


2.624 


3.774 


3.280 


49 


IS 


0.752 


0.659 


1.504 


1.319 


2.256 1.978 


3.007 


2.637 


3.759 


3.297 


45 


30 


0.749 


0.663 


1.498 


1.325 


2.247 1.988 


2.996 


2.650 


3.745 


3.313 


30 


45 


0.746 


0.666 


1.492 


1.332 


2.238 1.998 


2.984 


2.664 


3.730 


3.329 


15 


42 


0.743 


0.669 


1.486 


1.338 


2.229 2.007 


2.973 


2.677 


3.716 


3.346 


48 


15 


0.740 


0.672 


1.480 


1.345 


2.221 2.017 


2.961 


2.689 


3.701 


3.362 


45 


30 


0.737 


0.676 


1.475 


1.351 


2.212 2.027 


2.949 


2.702 


3.686 


3.378 


30 


45 


0.734 


0.679 


1.469 


1.358 


2.203 2.036 


2.937 


2.715 


3.672 


3.394 


15 


43 


0.731 


0.682 


1.463 


1.364 


2.194 2.046 


2.925 


2.728 


3.657 


3.410 


47 


15 


0.728 


0.685 


1.457 


1.370 


2.185 2.056 


2.913 


2.741 


3.642 


3.426 


45 


30 


0.725 


0.688 


1.451 


1.377 


2.176 2.065 


2.901 


2.753 


3.627 


3.442 


30 


45 


0.722 


0.692 


1.445 


1.383 


2.167 2.075 


2.889 


2.766 


3.612 


3.458 


15 


44 


0.719 


0.695 


1.439 


1.389 


2.158 2.084 


2.877 


2.779 


3.597 


3.473 


46 


15 


0.716 


0.698 


1.433 


1.396 


2.149 2.093 


2.865 


2.791 


3.582 


3.489 


45 


30 


0.713 


0.701 


1.427 


1.402 


2.140 2.103 


2.853 


2.804 


3.566 


3.505 


30 


45 


0.710 


0.704 


1.420 


1.408 


2.131 2.112 


2.841 


2.816 


3.551 


3.520 


15 


45 


0.707 


0.707 


1.414 


1.414 


2,121 2.121 


2.828 


2.828 


3.536 


3.536 


45 


o t 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


o r 


Bearing. 


Distance 1. 


Distance 2. 


Distance 3. 


Distance 4. 


Distance 5. 


Bearing. 



46° -60' 













30°- 


-45 


o 








75 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 


o f 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


O f 


30 15 


5.183 


3.023 


6.047 


3.526 


6.911 


4.030 


7.775 


4.534 


8.638 


5.038 


59 45 


30 


5.170 


3.045 


6.031 


3.553 


6.893 


4.060 


7.755 


4.568 


8.616 


5.075 


30 


45 


5.156 


3.068 


6.016 


3.579 


6.875 


4.090 


7.735 


4.602 


8.594 


5.113 


15 


31 


5.143 


3.090 


6.000 


3.605 


6.857 


4.120 


7.715 


4.635 


8.572 


5.150 


59 


15 


5.129 


3.113 


5.984 


3.631 


6.839 


4.150 


7.694 


4.669 


8.549 


5.188 


45 


30 


5.116 


3.135 


5.968 


3.657 


6.821 


4.180 


7.674 


4.702 


8.526 


5.225 


30 


45 


5.102 


3.157 


5.952 


3.683 


6.803 


4.210 


7.653 


4.736 


8.504 


5.262 


15 


32 


5.088 


3.180 


5.936 


3.709 


6.784 


4.239 


7.632 


4.769 


8.481 


5.299 


58 


15 


5.074 


3.202 


5.920 


3.735 


6.766 


4.269 


7.612 


4.802 


8.457 


5.336 


45 


30 


5.060 


3.224 


5.904 


3.761 


6.747 


4.298 


7.591 


4.836 


8.434 


5.373 


30 


45 


5.046 


3.246 


5.887 


3.787 


6.728 


4.328 


7.569 


4.869 


8.410 


5.410 


15 


33 


5.032 


3.268 


5.871 


3.812 


6.709 


4.357 


7.548 


4.902 


8.387 


5.446 


57 


15 


5.018 


3.290 


5.854 


3.838 


6.690 


4.386 


7.527 


4.935 


8.363 


5.483 


45 


30 


5.003 


3.312 


5.837 


3.864 


6.671 


4.416 


7.505 


4.967 


8.339 


5.519 


30 


45 


4.989 


3.333 


5.820 


3.889 


6.652 


4.445 


7.483 


5.000 


8.315 


5.556 


15 


34 


4.974 


3.355 


5.803 


3.914 


6.632 


4.474 


7.461 


5.033 


8.290 


5.592 


56 


15 


4.960 


3.377 


5.786 


3.940 


6.613 


4.502 


7.439 


5.065 


8.266 


5.628 


45 


30 


4.945 


3.398 


5.769 


3.965 


6.593 


4.531 


7.417 


5.098 


8.241 


5.664 


30 


45 


4.930 


3.420 


5.752 


3.990 


6.573 


4.560 


7.395 


5.130 


8.217 


5.700 


15 


35 


4.915 


3.441 


5.734 


4.015 


6.553 


4.589 


7.372 


5.162 


8.192 


5.736 


55 


15 


4.900 


3.463 


5.716 


4.040 


6.533 


4.617 


7.350 


5.194 


8.166 


5.772 


45 


30 


4.885 


3.484 


5.699 


4.065 


6.513 


4.646 


7.327 


5.226 


8.141 


5.807 


30 


45 


4.869 


3.505 


5.681 


4.090 


6.493 


4.674 


7.304 


5.258 


8.116 


5.843 


15 


36 


4.854 


3.527 


5.663 


4.115 


6.472 


4.702 


7.281 


5.290 


8.090 


5.878 


54 


15 


4.839 


3.548 


5.645 


4.139 


6.452 


4.730 


7.258 


5.322 


8.064 


5.913 


45 


30 


4.823 


3.569 


5.627 


4.164 


6.431 


4.759 


7.235 


5.353 


8.039 


5.948 


30 


45 


4.808 


3.590 


5.609 


4.188 


6.410 


4.787 


7.211 


5.385 


8.013 


5.983 


15 


37 


4.792 


3.611 


5.590 


4.213 


6.389 


4.815 


7.188 


5.416 


7.986 


6.018 


53 


15 


4.776 


3.632 


5.572 


4.237 


6.368 


4.842 


7.164 


5.448 


7.960 


6.053 


45 


30 


4.760 


3.653 


5.554 


4.261 


6.347 


4.870 


7.140 


5.479 


7.934 


6.088 


30 


45 


4.744 


3.673 


5.535 


4.286 


6.326 


4.898 


7.116 


5.510 


7.907 


6.122 


15 


38 


4.728 


3.694 


5.516 


4.310 


6.304 


4.925 


7.092 


5.541 


7.880 


6.157 


52 


15 


4.712 


3.715 


5.497 


4.334 


6.283 


4.953 


7.068 


5.572 


7.853 


6.191 


45 


30 


4.696 


3.735 


5.478 


4.358 


6.261 


4.980 


7.043 


5.603 


7.826 


6.225 


30 


45 


4.679 


3.756 


5.459 


4.381 


6.239 


5.007 


7.019 


5.633 


7.799 


6.259 


15 


39 


4.663 


3.776 


5.440 


4.405 


6.217 


5.035 


6.994 


5.664 


7.772 


6.293 


51 


15 


4.646 


3.796 


5.421 


4.429 


6.195 


5.062 


6.970 


5.694 


7.744 


6.327 


45 


30 


4.630 


3.816 


5.401 


4.453 


6.173 


5.089 


6.945 


5.725 


7.716 


6.361 


30 


45 


4.613 


3.837 


5.382 


4.476 


6.151 


5.116 


6.920 


5.755 


7.688 


6.394 


15 


40 


4.596 


3.857 


5.362 


4.500 


6.128 


5.142 


6.894 


5.785 


7.660 


6.428 


50 


15 


4.579 


3.877 


5.343 


4.523 


6.106 


5.169 


6.869 


5.815 


7.632 


6.461 


45 


30 


4.562 


3.897 


5.323 


4.546 


6.083 


5.196 


6.844 


5.845 


7.604 


6.495 


30 


45 


4.545 


3.917 


5.303 


4.569 


6.061 


5.222 


6.818 


5.875 


7.576 


6.528 


15 


41 


4.528 


3.936 


5.283 


4.592 


6.038 


5.248 


6.792 


5.905 


7.547 


6.561 


49 


15 


4.511 


3.956 


5.263 


4.615 


6.015 


5.275 


6.767 


5.934 


7.518 


6.594 


45 


30 


4.494 


3.976 


5.243 


4.638 


5.992 


5.301 


6.741 


5.964 


7.490 


6.626 


30 


45 


4.476 


3.995 


5.222 


4.661 


5.968 


5.327 


6.715 


5.993 


7.461 


6.659 


15 


43 


4.459 


4.015 


5.202 


4.684 


5.945 


5.353 


6.688 


6.022 


7.431 


6.691 


48 


15 


4.441 


4.034 


5.182 


4.707 


5.922 


5.379 


6.662 


6.051 


7.402 


6.724 


45 


30 


4.424 


4.054 


5.161 


4.729 


5.898 


5.405 


6.635 


6.080 


7.373 


6.756 


30 


45 


4.406 


4.073 


5.140 


4.752 


5.875 


5.430 


6.609 


6.109 


7.343 


6.788 


15 


43 


4.388 


4.092 


5.119 


4.774 


5.851 


5.456 


6.582 


6.138 


7.314 


6.820 


47 


15 


4.370 


4.111 


5.099 


4.796 


5.827 


5.481 


6.555 


6.167 


7.284 


6.852 


45 


30 


4.352 


4.130 


5.078 


4.818 


5.803 


5.507 


6.528 


6.195 


7.254 


6.884 


30 


45 


4.334 


4.149 


5.057 


4.841 


5.779 


5.532 


6.501 


6.224 


7.224 


6.915 


15 


44 


4.316 


4.168 


5.035 


4.863 


5.755 


5.557 


6.474 


6.252 


7.193 


6.947 


46 


15 


4.298 


4.187 


5.014 


4.885 


5.730 


5.582 


6.447 


6.280 


7.163 


6.978 


45 


30 


4.280 


4.206 


4.993 


4.906 


5.706 


5.607 


6.419 


6.308 


7.133 


7.009 


30 


45 


4.261 


4.224 


4.971 


4.928 


5.681 


5.632 


6.392 


6.336 


7.102 


7.040 


15 


45 


4.243 


4.243 


4.950 


4.950 


5.657 


5.657 


6.364 


6.364 


7.071 


7.071 


45 


o f 


Dap. 


Lat. 


Bep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


O f 


Bearing. 


Distance 6. 


Distance 7. 


Distance 8. 


Distance 9. 


Distance 10. 


Bearing. 



46° -60^ 



SUEVEYIN(t 



AND 



TRAVERSE TABLE 



G. A. WENTWOKTH, A.M. 

AUTHOR OF A SERIES OF TEXT-BOOKS -IN MATHEMATICS 



REVISED EDITION 



Boston, U.S.A., ani) London 
GINN^ & COMPANY, PUBLISHEKS 

1 8 9 G 



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